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Let $K$ be a field with a discrete valuation, and let $p$ be a prime. It isknown that if $\Gamma \lhd \Gamma_0 < \mathrm{PGL}_2(K)$ is a Schottky groupnormally contained in a larger group which is generated by order-$p$ elementseach fixing $2$ points $a_i, b_i \in \mathbb{P}_K^1$, then the quotient of acertain subset of the projective line $\mathbb{P}_K^1$ by the action of$\Gamma$ can be algebraized as a superelliptic curve $C : y^p = f(x) / K$. Thesubset $S \subset \mathbb{P}_K^1$ consisting of these pairs $a_i, b_i$ of fixedpoints is mapped modulo $\Gamma$ to the set of branch points of thesuperelliptic map $x : C \to \mathbb{P}_K^1$. We produce an algorithm fordetermining whether an input even-cardinality subset $S \subset \mathbb{P}_K^1$consists of fixed points of generators of such a group $\Gamma_0$ and which, inthe case of a positive answer, modifies $S$ into a subset $S^{\mathrm{min}}\subset \mathbb{P}_K^1$ with particularly nice properties. Our results do notinvolve any restrictions on the prime $p$ or on the residue characteristic of$K$ and allow these to be the same.

Branch points of split degenerate superelliptic curves I: non-archimedean uniformization