Abstract: Interplay between differential geometry and integrable systems is in most cases an one-way street: people apply differential- >geometric methods to study integrable systems. In my talk, I will lead you in the other direction: My main example will be the so called >c-projectively equivalent metrics. These are two Kähler metrics on one n-dimensional complex manifold $(M, J)$ such that their J-planar >curves, i.e., real-1-dimensional curves such that the acceleration is complex-proportional to the velocity, coincide. I will show that the >existence of a metric $g'$ that is c-projectively equivalent to $g$ allows us to construct, canonically, by invariant formulas, $2n$ integrals >for the geodesic flow of $g$. $n$ of the integrals are quadratic in velocities and $n$ of the integrals are linear in velocities. They are in >the involution so in the most non degenerate case we obtain the Liouville integrability. As the first application of the integrals, I will show >that their existence implies a very regular behaviour of the (1,1)-tensor $g^{-1}g'$: its Jordan block structure must be the same at all >points. Then, I use the integrals to give a local classification of c-projectively equivalent metrics. The Riemannian version of the >classification is due to Apostolov, Calderbank, Gauduchon, Tonnesen-Friedman. I will also slightly generalize it and obtain the Kähler->Liouville integrable systems of Kiyohara. We will see that the moduli space of the integrable systems coming from c-projectively >equivalent metrics is huge -- we have $n$ functional parameters that could be chosen almost freely. The last part of the talk is the proof >of the classical Yano-Obata conjecture in the c-projective theory -- integrals of the geodesic flow play the key role in this proof. If the time >allows, I will try to explain why, for certain setups in Riemannian or pseudo-Riemannian differential geometry, one can hope for and >almost expect the existence of integrals.