# M\"obius automorphisms of surfaces with many circles

## TL;DR

This paper classifies certain toric surfaces with specific circle-rich orbits under conformal transformations, extending the classical theory of Dupin cyclides by identifying conditions for their M"obius automorphism groups.

## Contribution

It generalizes the classification of Dupin cyclides to a broader class of toric surfaces with particular conformal orbit properties.

## Key findings

- Surfaces are toric with M"obius automorphism groups of dimension ≥ 2.
- Classifies real two-dimensional conformal orbits containing two circular arcs.
- Extends classical Dupin cyclide classification to new surface families.

## Abstract

We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a M\"obius automorphism group of dimension at least two. Our theorem generalizes the classical classification of Dupin cyclides.