{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T13:03:45Z","timestamp":1778591025690,"version":"3.51.4"},"reference-count":135,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,6,21]],"date-time":"2018-06-21T00:00:00Z","timestamp":1529539200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["3160121"],"award-info":[{"award-number":["3160121"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The purpose of the current article is to present a brief albeit accurate presentation of the main tools used in the study of symmetries of Lagrange equations for holonomic systems and subsequently to show how these tools are applied in the major models of modern cosmology in order to derive exact solutions and deal with the problem of dark matter\/energy. The key role in this approach are the first integrals of the field equations. We start with the Lie point symmetries and the first integrals defined by them, that is, the Hojman integrals. Subsequently, we discuss the Noether point symmetries and the well-known method for deriving the Noether integrals. By means of the Inverse Noether Theorem, we show that, to every Hojman quadratic first integral, it is possible to associate a Noether symmetry whose Noether integral is the original Hojman integral. It is emphasized that the point transformation generating this Noether symmetry need not coincide with the point transformation defining the Lie symmetry which produces the Hojman integral. We discuss the close connection between the Lie point and the Noether point symmetries with the collineations of the metric defined by the kinetic energy of the Lagrangian. In particular, the generators of Noether point symmetries are elements of the homothetic algebra of that metric. The key point in the current study of cosmological models is the introduction of the mini superspace, which is the space that is defined by the physical variables of the model, which is not the spacetime where the model evolves. The metric in the mini superspace is found from the kinematic part of the Lagrangian and we call it the kinetic metric. The rest part of the Lagrangian is the effective potential. We consider coordinate transformations of the original mini superspace metric in order to bring it to a form where we know its collineations, that is, the Killing vectors, the homothetic vector, etc. Then, we write the field equations of the cosmological model and we use the connection of these equations with the collineations of the mini superspace metric to compute the first integrals and subsequently to obtain analytic solutions for various allowable potentials and finally draw conclusions about the problem of dark energy. We consider the \u039bCDM cosmological model, the scalar field cosmology, the Brans\u2013Dicke cosmology, the f(R) gravity, the two scalar fields cosmology with interacting scalar fields and the Galilean cosmology. In each case, we present the relevant results in the form of tables for easy reference. Finally, we discuss briefly the higher order symmetries (the contact symmetries) and show how they are applied in the cases of scalar field cosmology and in the f(R) gravity.<\/jats:p>","DOI":"10.3390\/sym10070233","type":"journal-article","created":{"date-parts":[[2018,6,22]],"date-time":"2018-06-22T02:46:21Z","timestamp":1529635581000},"page":"233","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":89,"title":["Symmetries of Differential Equations in Cosmology"],"prefix":"10.3390","volume":"10","author":[{"given":"Michael","family":"Tsamparlis","sequence":"first","affiliation":[{"name":"Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics, University of Athens, Panepistemiopolis, 157 83 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9966-5517","authenticated-orcid":false,"given":"Andronikos","family":"Paliathanasis","sequence":"additional","affiliation":[{"name":"Instituto de Ciencias F\u00edsicas y Matem\u00e1ticas, Universidad Austral de Chile, Valdivia 5090000, Chile"},{"name":"Department of Mathematics and Natural Sciences, Core Curriculum Program, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia"},{"name":"Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hall, G.S. (2004). Symmetries and Curvature Structure in General Relativity, World Scientific. World Scientific Lecture Notes in Physics 46.","DOI":"10.1142\/1729"},{"key":"ref_2","unstructured":"Lie, S. (1888). Theorie der Transformationsgruppen I, B. G. Teubner."},{"key":"ref_3","unstructured":"Lie, S. (1888). Theorie der Transformationsgruppen II, B. G. Teubner."},{"key":"ref_4","unstructured":"Lie, S. (1888). Theorie der Transformationsgruppen III, B. G. Teubner."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ovsiannikov, L.V. (1982). Group Analysis of Differential Equations, Academic Press.","DOI":"10.1016\/B978-0-12-531680-4.50012-5"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Bluman, G.W., and Kumei, S. (1989). Symmetries and Differential Equations, Springer.","DOI":"10.1007\/978-1-4757-4307-4"},{"key":"ref_7","unstructured":"Ibragimov, N.H. (2000). CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws, CRS Press LLC."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Olver, P.J. (1993). Applications of Lie Groups to Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-4350-2"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0034-4877(84)90069-7","article-title":"Hidden symmetries and Killing tensors","volume":"20","author":"Crampin","year":"1984","journal-title":"Rep. Math. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2077","DOI":"10.1088\/0305-4470\/15\/7\/018","article-title":"Dynamical noether symmetries","volume":"15","author":"Kalotas","year":"1982","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1088\/0305-4470\/14\/3\/009","article-title":"On the Lie symmetries of the classical Kepler problem","volume":"14","author":"Prince","year":"1981","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1088\/0305-4470\/11\/2\/005","article-title":"Symmetry groups and conserved quantities for the harmonic oscillator","volume":"11","author":"Lutzky","year":"1978","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5349","DOI":"10.1088\/0305-4470\/28\/18\/023","article-title":"On the determination of non-local symmetries","volume":"28","author":"Govinder","year":"1995","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1137\/1023098","article-title":"Generalizations of Noether\u2019s theorem in classical mechanics","volume":"23","author":"Sarlet","year":"1981","journal-title":"SIAM Rev."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1995","DOI":"10.1002\/mma.934","article-title":"Symmetry group classification of ordinary differential equations: Survey of some results","volume":"30","author":"Mahomed","year":"2007","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/S0895-7177(97)00068-X","article-title":"The role of symmetries in solving differential equations","volume":"25","author":"Nucci","year":"1997","journal-title":"Math. Comp. Mod."},{"key":"ref_17","first-page":"001","article-title":"The differential form method for finding symmetries","volume":"1","author":"Harrison","year":"2005","journal-title":"Sigma"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5597","DOI":"10.1088\/0305-4470\/25\/21\/018","article-title":"Hidden symmetries associated with the projective group of nonlinear first-order ordinary differential equations","volume":"25","author":"Guo","year":"1992","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"301","DOI":"10.2991\/jnmp.1995.2.3-4.10","article-title":"On Lie reduction of the Navier-Stokes equations","volume":"2","author":"Popovych","year":"1995","journal-title":"Nonlinear Math. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1878","DOI":"10.1063\/1.525160","article-title":"Geodesic first integrals with explicit path-parameter dependence in Riemannian space-times","volume":"22","author":"Katzin","year":"1981","journal-title":"J. Math. Phys."},{"key":"ref_21","unstructured":"Noether, E. (1918). Invariante variationsprobleme, Nachr. d. K\u00f6nig. Gesellsch. d. Wiss. zu G\u00f6ttingen. Math.-Phys. Klasse, 235\u2013257."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Singer, S.F. (2004). Symmetry in Mechanics, Birkhauser Boston.","DOI":"10.1007\/978-1-4612-0189-2"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Costa, G., and Fogli, G. (2012). Symmetry and Group Theory in Particle Physics, Springer. Lecture Notes in Physics.","DOI":"10.1007\/978-3-642-15482-9"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Sundermeyer, K. (2014). Symmetries in Fundamental Physics, Springer.","DOI":"10.1007\/978-94-007-7642-5"},{"key":"ref_25","unstructured":"Witten, L. (1962). Conservation Laws in General Relativity, in Gravitation, and Introduction to Current Research, Willey."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s11071-010-9710-x","article-title":"Lie symmetries of geodesic equations and projective collineations","volume":"62","author":"Tsamparlis","year":"2010","journal-title":"Nonlinear Dyn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2957","DOI":"10.1007\/s10714-010-1054-9","article-title":"Lie and Noether symmetries of geodesic equations and collineations","volume":"42","author":"Tsamparlis","year":"2010","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2443","DOI":"10.1016\/j.geomphys.2012.09.004","article-title":"Lie point symmetries of a general class of PDEs: The heat equation","volume":"62","author":"Paliathanasis","year":"2012","journal-title":"J. Geom. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.geomphys.2016.05.004","article-title":"Lie and Noether point symmetries of a class of quasilinear systems of second-order differential equations","volume":"107","author":"Paliathanasis","year":"2016","journal-title":"J. Geom. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"175202","DOI":"10.1088\/1751-8113\/44\/17\/175202","article-title":"Two-dimensional dynamical systems which admit Lie and Noether symmetries","volume":"44","author":"Tsamparlis","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"L291","DOI":"10.1088\/0305-4470\/25\/7\/002","article-title":"A new conservation law constructed without using either Lagrangians or Hamiltonians","volume":"25","author":"Hojman","year":"1992","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/0020-7462(73)90039-5","article-title":"A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians","volume":"8","author":"Djukic","year":"1973","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"110","DOI":"10.2991\/jnmp.2002.9.s2.10","article-title":"Jacobi\u2019s last multiplier and the complete symmetry group of the Euler\u2013Poinsot system","volume":"9","author":"Nucci","year":"2002","journal-title":"J. Non. Math. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"065011","DOI":"10.1088\/0031-8949\/78\/06\/065011","article-title":"The Jacobi Last Multiplier and its applications in mechanics","volume":"78","author":"Nucci","year":"2008","journal-title":"Phys. Scr."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1142\/S1402925110000696","article-title":"Lagrangians for dissipative nonlinear oscillators: The method of Jacobi last multiplier","volume":"17","author":"Nucci","year":"2010","journal-title":"J. Non. Math. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2770","DOI":"10.1063\/1.528511","article-title":"Lie algebras associated with scalar second-order ordinary differential equations","volume":"30","author":"Mahomed","year":"1989","journal-title":"J. Math. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/0022-247X(90)90244-A","article-title":"Symmetry Lie algebras of nth order ordinary differential equations","volume":"151","author":"Mahomed","year":"1990","journal-title":"J. Math. Anal. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1088\/0305-4470\/9\/4\/007","article-title":"The Lie group of Newton\u2019s and Lagrange\u2019s equations for the harmonic oscillator","volume":"9","author":"Wulfman","year":"1976","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1119\/1.1986202","article-title":"Conservation laws for gauge-variant Lagrangians in classical mechanics","volume":"39","year":"1971","journal-title":"Am. J. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1017\/S1446788700007254","article-title":"The lengthening pendulum","volume":"9","author":"Werner","year":"1969","journal-title":"Aust. Math. Soc."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1103\/PhysRevLett.18.510","article-title":"Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians","volume":"18","author":"Lewis","year":"1967","journal-title":"Phys. Rev. Lett."},{"key":"ref_42","first-page":"1","article-title":"Second order differential equations. Conditions of complete integrability","volume":"9","author":"Ermakov","year":"1880","journal-title":"Universita Izvestia Kiev"},{"key":"ref_43","first-page":"681","article-title":"The nonlinear differential equation y\u2033 + p(x)y + cy\u22123 = 0","volume":"1","author":"Pinney","year":"1950","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2625","DOI":"10.1088\/0305-4470\/26\/11\/012","article-title":"On (2 + 1)-dimensional Ermakov systems","volume":"26","author":"Rogers","year":"1993","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1088\/0305-4470\/29\/4\/017","article-title":"Ermakov systems of arbitrary order and dimension: Structure and linearization","volume":"29","author":"Schief","year":"1996","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"146","DOI":"10.2298\/AADM0802146L","article-title":"The Ermakov equation: A commentary","volume":"2","author":"Leach","year":"2008","journal-title":"Appl. Anal. Discrete Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"275202","DOI":"10.1088\/1751-8113\/45\/27\/275202","article-title":"Generalizing the autonomous Kepler\u2013Ermakov system in a Riemannian space. A note on the construction of the Ermakov\u2013Lewis invariant","volume":"45","author":"Tsamparlis","year":"2012","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"5333","DOI":"10.1088\/0305-4470\/35\/25\/312","article-title":"A note on the construction of the Ermakov\u2013Lewis invariant","volume":"35","author":"Moyo","year":"2002","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1063\/1.1664886","article-title":"Curvature Collineations: A Fundamental Symmetry Property of the Space-Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor","volume":"10","author":"Katzin","year":"1969","journal-title":"J. Math. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1023\/A:1018827131768","article-title":"Some remarks on special conformal and special projective symmetries in general relativity","volume":"29","author":"Hall","year":"1997","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"082901","DOI":"10.1063\/1.4998715","article-title":"Lie and Noether point symmetries for a class of nonautonomous dynamical systems","volume":"58","author":"Karpathopoulos","year":"2017","journal-title":"Math. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"095202","DOI":"10.1088\/1751-8113\/47\/9\/095202","article-title":"Lie point and variational symmetries in minisuperspace Einstein gravity","volume":"47","author":"Christodoulakis","year":"2014","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_53","first-page":"183","article-title":"On the symmetries and invariants of the harmonic oscillator","volume":"19","author":"Gordon","year":"1986","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1103\/RevModPhys.61.1","article-title":"The cosmological constant problem","volume":"61","author":"Weinberg","year":"1989","journal-title":"Rev. Mod. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1103\/RevModPhys.75.559","article-title":"The cosmological constant and dark energy","volume":"75","author":"Peebles","year":"2003","journal-title":"Rev. Mod. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/S0370-1573(03)00120-0","article-title":"Cosmological constant\u2014The weight of the vacuum","volume":"380","author":"Padmanabhan","year":"2003","journal-title":"Phys. Rept."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"3406","DOI":"10.1103\/PhysRevD.37.3406","article-title":"Cosmological consequences of a rolling homogeneous scalar field","volume":"37","author":"Ratra","year":"1988","journal-title":"Phys. Rev D"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"043506","DOI":"10.1103\/PhysRevD.58.043506","article-title":"Evolution of the scale factor with a variable cosmological term","volume":"58","author":"Overduin","year":"1998","journal-title":"Phys. Rev. D"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"3511","DOI":"10.1103\/PhysRevD.80.083511","article-title":"Hubble expansion and structure formation in time varying vacuum models","volume":"80","author":"Basilakos","year":"2009","journal-title":"Phys. Rev. D"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/0370-2693(80)90670-X","article-title":"A new type of isotropic cosmological models without singularity","volume":"91","author":"Starobinsky","year":"1980","journal-title":"Phys. Lett. B"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"063509","DOI":"10.1103\/PhysRevD.67.063509","article-title":"Can the Chaplygin gas be a plausible model for dark energy?","volume":"67","author":"Gorini","year":"2003","journal-title":"Phys. Rev. D"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1088\/0264-9381\/10\/2\/009","article-title":"Scalar-field cosmologies","volume":"10","author":"Barrow","year":"1993","journal-title":"Class. Quantum Grav."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"127301","DOI":"10.1103\/PhysRevD.64.127301","article-title":"Scaling solutions and reconstruction of scalar field potentials","volume":"64","author":"Rubano","year":"2001","journal-title":"Phys. Rev. D"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/0375-9601(95)00123-K","article-title":"Matter creation and bulk viscosity in early cosmology","volume":"200","author":"Gariel","year":"1995","journal-title":"Phys. Lett. A"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2012.01.001","article-title":"Modified gravity and cosmology","volume":"513","author":"Clifton","year":"2012","journal-title":"Phys. Rep."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1146\/annurev-nucl-102115-044553","article-title":"Dark energy versus modified gravity","volume":"66","author":"Joyce","year":"2016","journal-title":"Annu. Rev. Nucl. Part. Sci."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"131302","DOI":"10.1103\/PhysRevLett.98.131302","article-title":"Are f(R)Dark Energy Models Cosmologically Viable?","volume":"98","author":"Amendola","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"044026","DOI":"10.1103\/PhysRevD.93.044026","article-title":"f(R) gravity as a dark energy fluid","volume":"93","author":"Battye","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"123510","DOI":"10.1103\/PhysRevD.88.123510","article-title":"Resembling dark energy and modified gravity with Finsler-Randers cosmology","volume":"88","author":"Basilakos","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.physletb.2011.09.082","article-title":"\u201cTeleparallel\u201d dark energy","volume":"704","author":"Geng","year":"2011","journal-title":"Phys. Lett. B"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"1091","DOI":"10.1103\/PhysRevD.42.1091","article-title":"New approach to find exact solutions for cosmological models with a scalar field","volume":"42","author":"Marmo","year":"1990","journal-title":"Phys. Rev. D."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"3412","DOI":"10.1063\/1.529455","article-title":"Killing tensors in two-dimensional space-times with applications to cosmology","volume":"32","author":"Rosquist","year":"1991","journal-title":"J. Math. Phys."},{"key":"ref_73","first-page":"268","article-title":"On the spaces of three dimensions that admit a continuous group of movements","volume":"11","author":"Bianchi","year":"1898","journal-title":"Soc. Ita. Mem. di Mat."},{"key":"ref_74","unstructured":"Rayan, M.P., and Shepley, L.C. (1975). Homogeneous Relativistic Cosmologies, Princeton University Press."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"217","DOI":"10.2307\/2370192","article-title":"Geometrical theorems on Einstein\u2019s cosmological equations","volume":"43","author":"Kasner","year":"1921","journal-title":"Am. J. Math."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"L277","DOI":"10.1088\/0305-4470\/14\/8\/004","article-title":"Exact solution for vacuum Bianchi type III model with a cosmological constant","volume":"14","author":"Moussiaux","year":"1981","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1088\/0264-9381\/24\/4\/008","article-title":"The general solution of Bianchi type III vacuum cosmology","volume":"24","author":"Christodoulakis","year":"2007","journal-title":"Class. Quantum Grav."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1007\/s10714-008-0678-5","article-title":"The general solution of Bianchi type VII h vacuum cosmology","volume":"41","author":"Terzis","year":"2009","journal-title":"Gen. Relat. Gravit."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"2734","DOI":"10.1103\/PhysRevD.15.2734","article-title":"Exact Bianchi IV cosmological model","volume":"15","author":"Harvey","year":"1977","journal-title":"Phys. Rev. D"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"1625","DOI":"10.1088\/0305-4470\/27\/5\/026","article-title":"Painlev\u00e9 analysis of the Mixmaster universe","volume":"27","author":"Cotsakis","year":"1994","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"2031","DOI":"10.1088\/0305-4470\/31\/8\/014","article-title":"On the integrability of Bianchi cosmological models","volume":"31","author":"Maciejewski","year":"1998","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_82","first-page":"0742901","article-title":"Integrability of the Bianchi IX system","volume":"46","author":"Libre","year":"2005","journal-title":"Math. Phys."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"022704","DOI":"10.1063\/1.2168123","article-title":"Formal and analytical integrability of the Bianchi IX system","volume":"47","author":"Libre","year":"2006","journal-title":"Math. Phys."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1088\/0305-4470\/28\/3\/019","article-title":"Non-integrability of the mixmaster universe","volume":"28","author":"Christiansen","year":"1995","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1007\/BF02722535","article-title":"N\u00f6ther\u2019s symmetries in fourth-order cosmologies","volume":"109","author":"Capozziello","year":"1994","journal-title":"Nuovo Cimento B"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"3259","DOI":"10.1088\/0264-9381\/14\/12\/011","article-title":"Conformal equivalence and Noether symmetries in cosmology","volume":"14","author":"Capozziello","year":"1997","journal-title":"Class. Quantum Grav."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"1615","DOI":"10.1023\/A:1001990303511","article-title":"Evolution of dynamical coupling in scalar tensor theory from Noether symmetry","volume":"32","author":"Modak","year":"2000","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0370-2693(01)01376-4","article-title":"Noether and some other dynamical symmetries in Kantowski-Sachs model","volume":"524","author":"Sanyal","year":"2002","journal-title":"Phys. Lett. B"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1142\/S0217751X02006080","article-title":"Exact solutions for cosmological models with a scalar field","volume":"17","author":"Motavali","year":"2002","journal-title":"IJMPA"},{"key":"ref_90","first-page":"676","article-title":"Beyond Einstein gravity: A Survey of gravitational theories for cosmology and astrophysics","volume":"36","author":"Kamilya","year":"2004","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s10714-006-0386-y","article-title":"Noether symmetry approach in matter-dominated cosmology with variable G and \u039b","volume":"39","author":"Bonanno","year":"2007","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"084023","DOI":"10.1103\/PhysRevD.76.084023","article-title":"Noether symmetries of Bianchi I, Bianchi III, and Kantowski-Sachs spacetimes in scalar-coupled gravity theories","volume":"76","author":"Camci","year":"2007","journal-title":"Phys. Rev. D"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"009","DOI":"10.1088\/1475-7516\/2007\/12\/009","article-title":"Reconstruction of the scalar\u2013tensor Lagrangian from a \u039bCDM background and Noether symmetry","volume":"12","author":"Capozziello","year":"2007","journal-title":"JCAP"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.physletb.2008.05.008","article-title":"Noether symmetry in f (R) cosmology","volume":"664","author":"Vakili","year":"2008","journal-title":"Phys. Lett. B"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1088\/1475-7516\/2008\/08\/016","article-title":"f (R) cosmology from Noether\u2019s symmetry","volume":"8","author":"Capozziello","year":"2008","journal-title":"JCAP"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.physletb.2008.04.061","article-title":"Dark energy and dust matter phases from an exact f (R)-cosmology model","volume":"664","author":"Capozziello","year":"2008","journal-title":"Phys. Lett. B"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"104030","DOI":"10.1103\/PhysRevD.80.104030","article-title":"Noether symmetry approach in phantom quintessence cosmology","volume":"80","author":"Capozziello","year":"2009","journal-title":"Phys. Rev. D"},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/j.physletb.2011.07.028","article-title":"Noether symmetry approach in f (R)-tachyon model","volume":"702","author":"Mubasher","year":"2011","journal-title":"Phys. Lett. B"},{"key":"ref_99","first-page":"314","article-title":"Symmetries of homogeneous cosmologies","volume":"4","author":"Cotsakis","year":"1998","journal-title":"Gravit. Cosmol."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.geomphys.2013.12.001","article-title":"FLRW metric f (R) cosmology with a perfect fluid by generating integrals of motion","volume":"77","author":"Dimakis","year":"2012","journal-title":"J. Geom. Phys."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"043528","DOI":"10.1103\/PhysRevD.93.043528","article-title":"Classical and quantum solutions in Brans-Dicke cosmology with a perfect fluid","volume":"93","author":"Paliathanasis","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"123535","DOI":"10.1103\/PhysRevD.91.123535","article-title":"Dynamical analysis in scalar field cosmology","volume":"91","author":"Paliathanasis","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s10714-015-2010-5","article-title":"Closed-form solutions of the Wheeler\u2013DeWitt equation in a scalar-vector field cosmological model by Lie symmetries","volume":"48","author":"Paliathanasis","year":"2016","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"064031","DOI":"10.1103\/PhysRevD.95.064031","article-title":"Noether symmetries and stability of ideal gas solutions in Galileon cosmology","volume":"95","author":"Dimakis","year":"2017","journal-title":"Phys. Rev. D"},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.geomphys.2016.11.022","article-title":"Exact solution of the Einstein-Skyrme model in a Kantowski-Sachs spacetime","volume":"114","author":"Paliathanasis","year":"2017","journal-title":"J. Geom. Phys."},{"key":"ref_106","unstructured":"Paliathanasis, A. (2014). Symmetries of Differential Equations and Applications in Relativistic Physics. [Ph.D. Thesis, University of Athens]."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1007\/s10509-015-2642-7","article-title":"Exact Scalar-Tensor Cosmological Solutions via Noether Symmetry","volume":"361","author":"Belinchon","year":"2016","journal-title":"Astrophys. Space Sci."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"044031","DOI":"10.1103\/PhysRevD.89.044031","article-title":"Minisuperspace canonical quantization of the Reissner-Nordstr\u00f6m black hole via conditional symmetries","volume":"89","author":"Christodoulakis","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"066","DOI":"10.1088\/1475-7516\/2016\/05\/066","article-title":"Conditional symmetries in axisymmetric quantum cosmologies with scalar fields and the fate of the classical singularities","volume":"16","author":"Zampeli","year":"2016","journal-title":"JCAP"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1142\/S021827189300026X","article-title":"Minisuperspace and Wheeler-DeWitt equation for string dilaton cosmology","volume":"2","author":"Capozziello","year":"1993","journal-title":"IJMPD"},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1023\/A:1001967102409","article-title":"Selection rules in minisuperspace quantum cosmology","volume":"32","author":"Capozziello","year":"2000","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1023\/A:1001935510837","article-title":"Higher-order corrections to the effective gravitational action from Noether symmetry approach","volume":"32","author":"Capozziello","year":"2000","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"1650183","DOI":"10.1142\/S0217732316501832","article-title":"Noether symmetries and duality transformations in cosmology","volume":"31","author":"Paliathanasis","year":"2016","journal-title":"MPLA"},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"1850093","DOI":"10.1142\/S0217732318500931","article-title":"Duality transformation and conformal equivalent scalar-tensor theories","volume":"33","author":"Gionti","year":"2018","journal-title":"MPLA"},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"123535","DOI":"10.1103\/PhysRevD.91.123535","article-title":"Dynamical analysis in scalar field cosmology","volume":"91","author":"Paliathanasis","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"1750206","DOI":"10.1142\/S0217732317502066","article-title":"Dust fluid component from Lie symmetries in scalar field cosmology","volume":"32","author":"Paliathanasis","year":"2017","journal-title":"MPLA"},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.physletb.2008.06.058","article-title":"Scalar-tensor cosmology with R- 1 curvature correction by Noether symmetry","volume":"666","author":"Motavali","year":"2008","journal-title":"Phys. Lett. B"},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"103512","DOI":"10.1103\/PhysRevD.83.103512","article-title":"Using the Noether symmetry approach to probe the nature of dark energy","volume":"83","author":"Basilakos","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"123514","DOI":"10.1103\/PhysRevD.84.123514","article-title":"Constraints and analytical solutions of theories of gravity using Noether symmetries","volume":"84","author":"Paliathanasis","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"015006","DOI":"10.1088\/0264-9381\/29\/1\/015006","article-title":"Three-fluid cosmological model using Lie and Noether symmetries","volume":"29","author":"Tsamparlis","year":"2012","journal-title":"Class. Quantum Grav."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"104042","DOI":"10.1103\/PhysRevD.89.104042","article-title":"New Schwarzschild-like solutions in f(T) gravity through Noether symmetries","volume":"89","author":"Paliathanasis","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"043529","DOI":"10.1103\/PhysRevD.90.043529","article-title":"Two scalar field cosmology: Conservation laws and exact solutions","volume":"90","author":"Paliathanasis","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1006\/jmaa.1993.1295","article-title":"Lie and Noether counting theorems for one-dimensional systems","volume":"178","author":"Mahomed","year":"1993","journal-title":"J. Math. An. Appl."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.physletb.2016.01.049","article-title":"On the Hojman conservation quantities in Cosmology","volume":"755","author":"Paliathanasis","year":"2016","journal-title":"Phys. Lett. B"},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"2893","DOI":"10.1142\/S0217732307025893","article-title":"Unified dark matter in scalar field cosmologies","volume":"22","author":"Bertacca","year":"2007","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1103\/PhysRev.124.925","article-title":"Mach\u2019s principle and a relativistic theory of gravitation","volume":"124","author":"Brans","year":"1961","journal-title":"Phys. Rev."},{"key":"ref_127","first-page":"114","article-title":"On solvable Lie algebras","volume":"32","author":"Mubarakzyanov","year":"1963","journal-title":"Izv. Vyss. Uchebn Zavendeni\u012d Mat."},{"key":"ref_128","first-page":"99","article-title":"Classification of real structures of Lie algebras of fifth order","volume":"34","author":"Mubarakzyanov","year":"1963","journal-title":"Izv. Vyss. Uchebn Zavendeni\u012d Mat."},{"key":"ref_129","first-page":"104","article-title":"Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element","volume":"35","author":"Mubarakzyanov","year":"1963","journal-title":"Izv. Vyss. Uchebn Zavendeni\u012d Mat."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1103\/RevModPhys.82.451","article-title":"f(R) theories of gravity","volume":"82","author":"Sotiriou","year":"2010","journal-title":"Rev. Mod. Phys."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"084003","DOI":"10.1103\/PhysRevD.79.084003","article-title":"Covariant galileon","volume":"79","author":"Deffayet","year":"2009","journal-title":"Phys. Rev. D"},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"103524","DOI":"10.1103\/PhysRevD.90.103524","article-title":"Dynamical symmetries and observational constraints in scalar field cosmology","volume":"10","author":"Paliathanasis","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_133","doi-asserted-by":"crossref","first-page":"075012","DOI":"10.1088\/0264-9381\/33\/7\/075012","article-title":"f (R)-gravity from Killing tensors","volume":"33","author":"Paliathanasis","year":"2016","journal-title":"Class. Quantum Gravit."},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"024021","DOI":"10.1103\/PhysRevD.95.024021","article-title":"Dynamical symmetries in Brans-Dicke cosmology","volume":"95","author":"Papagiannopoulos","year":"2017","journal-title":"Phys. Rev. D"},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1016\/j.physletb.2012.10.073","article-title":"Generalized Noether symmetry in f (T) gravity","volume":"718","author":"Sadjadi","year":"2012","journal-title":"Phys. Lett. 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