Generalized Fermat Riemann surfaces of infinite type

The Loch Ness monster (LNM) is, up to homeomorphisms, the unique orientable, connected, Hausdorff, second countable surface of infinite genus and with exactly one end. For each integer $k\ge 2$, we construct Riemann surface structures $S$ on the LNM admitting a group of conformal automorphisms $H\cong {\mathbb{Z}}_k^{\mathbb{N}}$ such that $S\mathrm{/}H$ is planar. These structures can be described algebraically inside the projective space ${\mathbb{P}}^{\mathbb{N}}$ after deleting some limit points.