  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 one 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, that is the
space which 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.
