On the Finiteness of the Number of Orbits on Quasihomogeneous $(\mathbb C^*)^k\times SL_2(\mathbb C)$-manifolds

We obtain an effective criterion for the finiteness of the number of orbits contained in the closure of a given $G$-orbit for the case of a rational linear action of the group $G:=(\mathbb C^*)^k\times SL_2(\mathbb C)$ on a finite-dimensional linear space as well as on the projectivization of such a space.