Recently, I read a paper about discrete Schrödinger operator. There is a map related to trace map from $C^3$ to $C^3$ as follows:
$$T(x,y,z)=(y,z,yz-x).$$
We can calculated that $T$ has the folliwng invariant surface $$x^2+y^2+z^2-xyz-2=D.$$
Of course in this case, we have some geometry explanations for it.
My question is that give a general diffeomorphism $T$ from $C^n$ to $C^n$, is it possible to given some criterion get invariant surface or get rid off invariant surface?