This paper deals with the solvability at \(\Sigma =\{0\}\times S^1\) (in sense of Hörmander) of first-order operators \(P=L-p\) , where p is a smooth function defined in \(\Omega _\epsilon =(-\epsilon ,\epsilon )\times S^1\) , \(\epsilon>0\) , and L is a complex vector field in the form \(L=\partial /\partial t+(x^na(x)+ix^{2n-1}b(x))\partial /\partial x\) , with \(a(0)\ne 0\) , \(b(0)\ne 0\) , and \(n\ge 2\) , defined on \(\Omega _\epsilon =(-\epsilon ,\epsilon )\times S^1\) , where a and b are real-valued smooth functions in \((-\epsilon ,\epsilon )\) . Given \(k\in \mathbb {N}\) , we present necessary and sufficient conditions to obtain solutions in \(C^k(\Omega _\epsilon )\) to the equation \(Lu=pu+f\) in a neighborhood of \(\Sigma\) . Also, we study equations in the form \(Lu=pu+q{\overline{u}}+f\) .
