INTRO:
If you don't feel like reading the justification for my question, just skip down to the question.
Some recurrence relations can be solved by finding invariants, or quantities that remain unchanged. For example, consider the sequences defined by $$a_{n+1}=\frac{a_nb_n}{a_n+b_n+1}$$ $$b_{n+1}=a_n+b_n$$ $$a_0=b_0=-1$$ If one defines the function $\mu$ as $$\mu (x,y)=x+xy+y$$ then it is easily proven that $$\mu (x,y)=\mu\bigg(\frac{xy}{x+y+1}, x+y\bigg)$$ From this, it follows that $$\mu (a_{n+1}, b_{n+1})=\mu (a_n,b_n)$$ and $$\mu (a_n,b_n)=\mu(a_0,b_0)$$ $$\mu (a_n,b_n)=-1$$ $$a_n+a_nb_n+b_n=-1$$ $$a_n=\frac{-1-b_n}{1+b_n}$$ $$a_n=-1$$ and so $$b_{n+1}=-1+b_n$$ $$b_{n+1}=b_n-1$$ ...yielding explicit formulas for $a_n$ and $b_n$: $$a_n=-1,\space\space\space b_n=-1-n$$
This was perhaps a trivial (and manufactured) example, but it demonstrates how invariants are used to solve recursions.
QUESTION:
I would like to know the following: what are some methods used to find invariants? That is, what methods might one use to find a particular nontrivial solution $\mu$ to the functional equation $$\mu(x,y)=\mu(f(x,y),g(x,y))$$ where $f,g$ are given?