Putnam 1987 A5

Let $$\vec{G}(x,y) = \left( \frac{-y}{x^2+4y^2}, \frac{x}{x^2+4y^2},0 \right).$$ Prove or disprove that there is a vector-valued function $$\vec{F}(x,y,z) = (M(x,y,z), N(x,y,z), P(x,y,z))$$ with the following properties:

(i) $M,N,P$ have continuous partial derivatives for all $(x,y,z) \neq (0,0,0)$;

(ii) $\mathrm{Curl}\,\vec{F} = \vec{0}$ for all $(x,y,z) \neq (0,0,0)$;

(iii) $\vec{F}(x,y,0) = \vec{G}(x,y)$.