  We discuss the approximation of the eigensolutions associated with the
Maxwell eigenvalues problem in the framework of least-squares finite elements.
We write the Maxwell curl curl equation as a system of two first order equation
and design a novel least-squares formulation whose minimum is attained at the
solution of the system. The eigensolution are then approximated by considering
the eigenmodes of the underlying solution operator. We study the convergence of
the finite element approximation and we show several numerical tests confirming
the good behavior of the method. It turns out that nodal elements can be
successfully employed for the approximation of our problem also in presence of
singular solutions.
