= Elliptically scaled angular one-form
{title2=$\omega_{a,b}=ab(x\,dy-y\,dx)/(a^2x^2+b^2y^2)$}
For real $a,b\ne0$, 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 $f=\operatorname{Arg}(ax+iby)$. In a chart with $x\ne0$, 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 $x=\cos t/a$, $y=\sin t/b$, $0\leq t\leq2\pi$, substitution gives $\omega_{a,b}=dt$, hence its <line integral> is $2\pi\ne0$. 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 $2\pi\operatorname{sgn}(ab)$. 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.
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