This paper deals with semi-global C k -solvability of complex vector fields of. It is shown that the interplay between the order of vanishing of the functions a and b at x = 0 influences the C k -solvability at = {0} × S 1 . When r = 1, it is permitted that the functions a and b of L depend on the x and t variables, that is, L = ∂/∂t + x(a(x, t) + ib(x, t))∂/∂ x, where (x, t) ∈ .