Prescribing analytic singularities for solutions of a class of vector fields on the torus

We consider the operator L = partial derivative t + ( a( t) + ib( t)) partial derivative(x) acting on distributions on the two-torus T-2, where a and b are real-valued, real analytic functions defined on the unit circle T-1. We prove, among other things, that when b changes sign, given any subset Sigma of the set of the local extrema of the local primitives of b, there exists a singular solution of L such that the t-projection of its analytic singular support is Sigma; furthermore, for any tau is an element of Sigma and any closed subset F of T-x(1) there exists u is an element of D'(T-2) such that Lu is an element of C-omega(T-2) and sing supp(A)(u) = {tau} x F. We also provide a microlocal result concerning the trace of u at t = tau.