Integrability of planar polynomial differential systems through linear differential equations

In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constructed by using this method. We also present a new example of this kind of families. We give an analogous method for constructing rational equations but by means of a linear differential equation of first order.Comment: 24 pages, no figure

Finiteness of integrable nn-dimensional homogeneous polynomial potentials

We consider natural Hamiltonian systems of $n>1$ degrees of freedom with polynomial homogeneous potentials of degree $k$. We show that under a genericity assumption, for a fixed $k$, at most only a finite number of such systems is integrable. We also explain how to find explicit forms of these integrable potentials for small $k$

Some Noncommutative Matrix Algebras Arising in the Bispectral Problem

I revisit the so called "bispectral problem" introduced in a joint paper with Hans Duistermaat a long time ago, allowing now for the differential operators to have matrix coefficients and for the eigenfunctions, and one of the eigenvalues, to be matrix valued too. In the last example we go beyond this and allow both eigenvalues to be matrix valued