I have an invertible linear transformation $T:F^k\to F^k$, where $F$ is a finite field and $k$ is a natural number. It's easy to find the linear subspaces S that are invariant under T. How do I find non-linear sets S (e.g., sets defined by degree-2 multivariate polynomials $\{ (p_1(t_1,...,t_n),p_2(t_1,...,t_n)...)|t_1,...,t_n \in F \}$ that are invariant under T?

The context in which this question arose in my research: I'm looking for a certain construction in which sets have to satisfy two properties: one is invariance under $T$, the other is a property that is pretty well understood. Any information about invariance of non-linear sets (e.g., forms of $T$ from which its non-linear invariant sets could be easily deduced, pointers to relevant math theorems, etc) might be useful for me.