Lie algebras of vertical derivations on semiaffine varieties with torus actions

Let X be a normal variety endowed with an algebraic torus action. An additive group action $\alpha$ on is called vertical if general orbit of contained in the closure and image normalizes Aut(X). Our first result this paper classification actions under assumption that proper over affine variety. Then we establish criterion as to when infinitesimal generators finite collection generate finite-dimensional Lie algebra inside derivations X.