  We study the existence of invariant quadrics for a class of systems of
difference equations in ${\mathbb R}^n$ 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.
