  We describe all metrics geodesically compatible with a 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.
