Invariant quadrics and orbits for a family of rational systems of difference equations

We study the existence of invariant quadrics for a class of systems of difference equations in Rn defined by linear fractionals sharing denominator. Such systems can be described in terms of some square matrix A and we prove that there is a correspondence between non-degenerate invariant quadrics and solutions to a certain matrix equation involving A. We show that if A is semisimple and the corresponding system admits non-degenerate quadrics, then every orbit of the dynamical system is contained either in an invariant affine variety or in an invariant quadric.