We reprove the fact, due to Backus, that the Poincaré operator in ellipsoids admits a pure point spectrum with polynomial eigenfunctions. We then show that the eigenvalues of the Poincaré operator restricted to polynomial vector fields of fixed degree admits a limit repartition given by a probability measure that we construct explicitely. For that, we use Fourier integral operators and ideas coming from Alan Weinstein and the first author in the seventies. The starting observation is that the orthogonal polynomials in ellipsoids satisfy a PDE.