  For a Veech surface (x,\omega), we characterize subspaces of X^n, invariant
under the diagonal action of the affine group of X. We prove that
non-arithmetic Veech surfaces have only finitely many invariant subspaces of
very particular shape (in any dimension). Among other consequences we find
copies of (X,\omega) embedded in the moduli-space of translation surfaces. We
study illumination problems in (pre-)lattice surfaces. For (X,\omega)
prelattice we prove the at most countableness of points non-illuminable from
any x in X. Applying our results on invariant subspaces we prove the finiteness
of these sets when (X,\omega) is Veech.
