Exact solutions to the partially integrable Eckhaus equation

A partially integrable extension of the Eckhaus equation is first converted to one real fourth order equation. The only integrable case is isolated by simply solving a diophantine equation, and its linearizing transformation, not obvious at first glance, is shown to be the singular part transformation of Painlevé analysis. In the partially integrable case, three exact solutions are found by the truncation procedure. The third one is a six-parameter solution, whose dependence on $x$ is elliptic and dependence on $t$ involves the equation of Chazy.