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