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We study the existence of invariant quadrics for a class of systems ofdifference equations in ${\mathbb R}^n$ defined by linear fractionals sharingdenominator. Such systems can be described in terms of some square matrix $A$and we prove that there is a correspondence between non-degenerate invariantquadrics and solutions to a certain matrix equation involving $A$. We show thatif $A$ is semisimple and the corresponding system admits non-degeneratequadrics, then every orbit of the dynamical system is contained either in aninvariant affine variety or in an invariant quadric.

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