Exact Solution of the Kompaneets Equation for Strong Explosion in Medium   with Inverse-Square Decreasing Density
  In the framework of the Kompaneets approximation the propagation of a shock front (SF) in inhomogeneous medium with the power-law density decrease at the exponent $n=2$ is investigated. For this important case corresponding to outer regions of the solar and stellar coronas an unexpectedly simple exact solution clearing a structure of the general solution for an arbitrary monotonic density of medium is found. Our results for plane-layered medium are compared to the Korycansky one for off-centre explosion in radially stratified medium. The relation between the solutions in these media are found and a new exact solution for a noncentral explosion in the case of density singularity on the finite radius is received. 
V.M.Kontorovich and S.F.Pimenov