  An interesting discovery in the last two years in the field of mathematical
physics has been the exceptional $X_\ell$ Laguerre and Jacobi polynomials.
Unlike the well-known classical orthogonal polynomials which start with
constant terms, these new polynomials have lowest degree $\ell=1,2,...$, and
yet they form complete set with respect to some positive-definite measure.
While the mathematical properties of these new $X_\ell$ polynomials deserve
further analysis, it is also of interest to see if they play any role in
physical systems. In this paper we indicate some physical models in which these
new polynomials appear as the main part of the eigenfunctions. The systems we
consider include the Dirac equations coupled minimally and non-minimally with
some external fields, and the Fokker-Planck equations. The systems presented
here have enlarged the number of exactly solvable physical systems known so
far.
