Integrable Systems of Algebraic Origin and Separation of Variables

A plane algebraic curve whose Newton polygon contains $d$ integer points is completely determined by $d$ points in the plane through which it passes. Its coefficients regarded as functions of sets of coordinates of these points commute with respect to the Poisson bracket corresponding to the pair of coordinates of any of these points. This observation was made by Babelon and Talon in 2002. A result more general in some respects and less general in others was obtained by Enriquez and Rubtsov in 2003. As a particular case, we obtain that the coefficients of the Lagrange interpolation polynomial commute with respect to a Poisson bracket on the set of interpolation data. We prove a general assertion in the framework of the method of separation of variables which explains all these facts. This assertion is as follows: Any (nondegenerate) system of $n$ smooth functions in $n+2$ variables generates an integrable system with $n$ degrees of freedom. In addition to those mentioned above, the examples include a version of the Hermite interpolation polynomial and systems related to Weierstrass models of curves (= miniversal deformations of singularities). The integrable system related to the Lagrange interpolation polynomial has recently arisen as a reduction of rank-2 Hitchin systems (and, thereby, it gives particular solutions of such systems; see the author's paper in Doklady Mathematics), and it is also closely related to the integrable systems on universal bundles of symmetric powers of curves introduced by Buchstaber and Mikhailov in 2017.