The existence of a solution to the linearized problem of the rotation of a viscous two-phase drop consisting of compressible and incompressible fluids is proved. The inner fluid is the incompressible one. It is bounded by a closed, unknown surface that does not intersect the outer free boundary. The compressible fluid is barotropic. Surface tension forces act on both boundaries. It is assumed that the angular velocity is small, and the drop shape is close to a two-layer equilibrium figure. A theorem on the global (in time) solvability of the problem is obtained in Sobolev–Slobodetski\v{\i} spaces.