Natural coordinate of a holomorphic differential ($d{w}^m=\varphi (z)d{z}^m$)

For a nonvanishing local $m$-differential $\varphi (z)d{z}^m$, choose $w=\int {\varphi }^{1/m}dz$. Then the differential is $d{w}^m$. Different root choices give $w\mapsto \zeta w+b$, ${\zeta }^m=1$. For $m=1,2$ the permitted linear parts are translations or signs, producing translation surfaces and half-translation surfaces.