Singularities of Separation Coordinates

We study singularities of separation coordinates for geodesic flows of real-analytic two-dimensional (pseudo-)Riemannian metrics. We obtain a complete local classification of these singularities using the Nijenhuis operator associated with a quadratic first integral. The singular set is locally a smooth curve, an isolated point, or two transversely intersecting curves. We also show that orthogonal separation near one point of an analytic disk extends to a quadratic first integral on the entire disk and provide explicit separation coordinates near singularities.