  A Lie system is a system of first-order ordinary differential equations
describing the integral curves of a $t$-dependent vector field taking values in
a finite-dimensional real Lie algebra of vector fields: a so-called
Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class
of Lie systems, the $k$-symplectic Lie systems, admitting a Vessiot-Guldberg
Lie algebra of Hamiltonian vector fields with respect to the presymplectic
forms of a $k$-symplectic structure. We devise new $k$-symplectic geometric
methods to study their superposition rules, time independent constants of
motion and general properties. Our results are illustrated by examples of
physical and mathematical interest. As a byproduct, we find a new interesting
setting of application of the $k$-symplectic geometry: systems of first-order
ordinary differential equations.
