Let $R$ be a ring generated by $k$ rational functions in the variables $x_1,...,x_n$ over the real numbers.

Is there an algorithm that computes a set of rational functions $f_1,...,f_l \in R$ which generate the subring $Q$ of $R$ consisting of all those rational functions in $R$ which are invariant under the transformation $(x_1,...,x_n) \rightarrow (-x_1,...,-x_n)$?

Background of the question: I have a birational map $F$ on n dimensional real affine space and I would like to find conserved quantities of the map, that is a function H such that $H \circ f = H$. Now, the map f is the composition of two involutions $I_1$, $I_2$, where $I_1$ is simply defined by $x \rightarrow -x$. The involution $I_2$ is a quite complicated map, but I am nevertheless able to find a set of rational functions generating the ring of all conserved quantities of $I_2$. Hence, I would like to know whether $I_1$ and $I_2$ have common conserved quantities, or in other words whether $I_2$ has conserved quantities which are invariant under $x \rightarrow -x$.