 Elliptically scaled angular one-form

For real $a,b\ne 0$, this differential one-form on the punctured plane is the pullback of a differential form of the nonexact angular one-form under the invertible linear transformation $(X,Y)=(ax,by)$. It is a closed differential form for every such pair. On a region admitting a continuous complex argument, it has the local potential $f=\mathrm{Arg}(ax+iby)$. In a chart with $x\ne 0$, this can be written $f=\arctan (by/(ax))+C$.

There is no globally single-valued potential on the whole punctured plane. On the closed path $x=\cos t/a$, $y=\sin t/b$, $0\le t\le 2\pi$, substitution gives ${\omega }_{a,b}=dt$, hence its line integral is $2\pi \ne 0$. This contradicts the zero integral around every closed path required of an exact differential form. Around a positively oriented once-winding path the integral is $2\pi \mathrm{sgn}(ab)$. The local equation ${\omega }_{a,b}=0$ gives radial rays; away from a vertical ray its solutions as graphs are $y=Cx$ on intervals not crossing the excluded origin.