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We obtain the solutions of the generic bilinear master equation for a quantumoscillator with constant coefficients in the Gaussian form. Thewell-behavedness and positive semidefiniteness of the stationary states couldbe characterized by a three-dimensional Minkowski vector. By requiring thestationary states to satisfy a factorized condition, we obtain a generic classof master equations that includes the well-known ones and theirgeneralizations, some of which are completely positive. A further subset of themaster equations with the Gibbs states as stationary states is also obtained.For master equations with not completely positive generators, an analysis onthe stationary states suggests conditions on the coefficients of the masterequations that generate positive evolution for a given initial state.

Solutions of generic bilinear master equations for a quantum oscillator -- positive and factorized conditions on stationary states