A classification of rotary embeddings of multicycles

We classify rotary (orientably-regular) maps whose underlying graphs are multicycles. For the multicycle ${\mathrm{C}}_n^{(\lambda )}$ of length $n$ and edge-multiplicity $\lambda$, we determine all rotary embeddings for $n⩾3$ and $\lambda ⩾2$. When $n$ is odd, there is a unique isomorphism class; when $n$ is even, the embeddings form a family ${\mathcal{M}}_n^{(\lambda )}(i,j)$ parameterized by integer pairs $(i,j)$ satisfying explicit congruence conditions.