  A {\it Lie system} is a nonautonomous system of first-order differential
equations admitting a {\it superposition rule}, i.e., a map expressing its
general solution in terms of a generic family of particular solutions and some
constants. Using that a Lie system can be considered as a curve in a
finite-dimensional Lie algebra of vector fields, a so-called {\it
Vessiot--Guldberg Lie algebra}, we associate every Lie system with a Lie
algebra of Lie point symmetries induced by the Vessiot--Guldberg Lie algebra.
This enables us to derive Lie symmetries of relevant physical systems described
by first- and higher-order systems of differential equations by means of Lie
systems in an easier way than by standard methods. A generalization of our
results to partial differential equations is introduced. Among other
applications, Lie symmetries for several new and known generalizations of the
real Riccati equation are studied.
