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A major drawback of most methods to find analytic expressions for solitary waves is the a priori restriction to a given class of expressions. To overcome this difficulty, we present a new method, applicable to a wide class of autonomous equations, which builds as an intermediate information the first order autonomous ODE satisfied by the solitary wave. We discuss its application to the cubic complex one-dimensional Ginzburg-Landau equation, and conclude to the elliptic nature of the yet unknown most general solitary wave.

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The perturbation of an exact solution exhibits a movable transcendental essential singularity, thus proving the nonintegrability. Then, all possible exact particular solutions which may be written in closed form are isolated with the perturbative Painlevé test; this proves the inexistence of any vacuum solution other than the three known ones.