Consider a linear algebraic group $G$ acting on an affine variety. I am interested in knowing some information about the following two questions:
1) Is there a subvariety whose image under $G$ is the whole variety?
2) If we have a subvariety which is locally transverse to the orbits, under which conditions we can ensure that its image under $G$ is the whole variety?
If the answers to these questions are not trivial, I greatly appreciate suggesting some reference.