We study the solvability of equations associated with a complex vector field $L$ in ${ℝ}^2$ with $C^{\infty }$ or $C^{\omega }$ coefficients. We assume that $L$ is elliptic everywhere except on a simple and closed curve $\Sigma$. We assume that, on $\Sigma$, $L$ is of infinite type and that $L\wedge \overline{L}$ vanishes to a constant order. The equations considered are of the form $Lu=pu+f$, with $f$ satisfying compatibility conditions. We prove, in particular, that when the order of vanishing of $L\wedge \overline{L}$ is $\>1$, the equation $Lu=f$ is solvable in the $C^{\infty }$ category but not in the $C^{\omega }$ category....