Quasihomogeneous $(C^*)^k \times SL_2(C)$-varieties containing a finite   number of orbits
  Let $G:= (C^*)^k\times SL_2(C)$ act linearly on a vector space or its projectivisation. We obtain an effective criterion to detect whether a number of orbits in an orbit-closure is finite or not. 
E. V. Sharoyko