Nonexistence of global solutions for a class of complex vector fields on two-torus

The goal of this paper is study the global solvability of a class of complex vector fields of the special form L=∂/∂t+(a+ib)(x)∂/∂x, a,b∈C∞(S1;R), defined on two-torus T2≅R2/2πZ2. The kernel of transpose operator is described and the solvability near the characteristic set is also studied.