Singular solutions of complex vector fields on the Möbius band

Let $M$ be the Möbius band, uniformized as $R^2/G$ where G is the automorphism group generated by the isometry $(x,y)\mapsto (x+1,-y)$. We show the existence of singular solutions u to the complex vector field$L\doteq \frac{\partial }{\partial x}+y\left(a\right(x,y)+ib(x,y\left)\right)\frac{\partial }{\partial y},a,b\in C^{\infty }(R^2;R)$ where both L and u pass to the quotient. In other words, the passage of L to the Möbius band is not globally hypoelliptic. Some consequences for the question of whether non-orientable surfaces admit globally hypoelliptic vector fields are examined.