  The $k$-symplectic structures appear in the geometric study of the partial
differential equations of classical field theories. Meanwhile, we present a new
application of the $k$-symplectic structures to investigate a type of systems
of first-order ordinary differential equations, the $k$-symplectic Lie systems.
In particular, we analyse the properties, e.g. the superposition rules, of a
new example of $k$-symplectic Lie system which occurs in the analysis of
diffusion equations.
