From quadratic integrals to Nijenhuis operators: two-dimensional dictionary

RESUMO

We construct an explicit dictionary between two local classifications in dimension two: normal forms of pseudo-Riemannian metrics with a quadratic integral of the geodesic flow and normal forms of $\operatorname{gl}$-regular Nijenhuis operators. The correspondence is obtained by an explicit change of coordinates constructed from the trace and determinant of the operator.