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

We consider the operator $L=\partial_t+(a(t)+ib(t))\partial_x$ acting on distributions on the two-torus $\mathbb T^2,$ where $a$ and $b$ are real-valued, real analytic functions defined on the unit circle $\mathbb T^1.$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 $\tau\in\Sigma$ and any closed subset $F$ of $\mathbb T^1_x$ there exists $u\in\mathcal D'(\mathbb T^2)$ such that $Lu\in C^\omega(\mathbb T^2)$ and $\operatorname{sing\, supp_A}(u)=\{\tau\}\times F.$ We also provide a microlocal result concerning the trace of $u$ at $t=\tau.$