  We study the Lie and Noether point symmetries of a class of systems of
second-order differential equations with $n$ independent and $m$ dependent
variables ($n\times m$ systems). We solve the symmetry conditions in a
geometric way and determine the general form of the symmetry vector and of the
Noetherian conservation laws. We prove that the point symmetries are generated
by the collineations of two (pseudo)metrics, which are defined in the spaces of
independent and dependent variables. We demonstrate the general results in two
special cases (a) a system of $m$ coupled Laplace equations and (b) the
Klein-Gordon equation of a particle in the context of Generalized Uncertainty
Principle. In the second case we determine the complete invariant group of
point transformations, and we apply the Lie invariants in order to find
invariant solutions of the wave function for a spin-$0$ particle in the two
dimensional hyperbolic space.
