Applications of Nijenhuis Geometry V: Geodesic Equivalence and Finite-Dimensional Reductions of Integrable Quasilinear Systems

We describe all metrics geodesically compatible with a \(\textrm{gl}\)-regular Nijenhuis operator L. The set of such metrics is large enough so that a generic local curve \(\gamma \) is a geodesic for a suitable metric g from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from L preserves the property of \(\gamma \) to be a g-geodesic. This implies that every metric g geodesically compatible with L gives us a finite-dimensional reduction of this PDE system. We show that its restriction onto the set of g-geodesics is naturally equivalent to the Poisson action of \(\mathbb {R}^n\) on the cotangent bundle generated by the integrals coming from geodesic compatibility.