The eigenvalue problem of an infinite-dimensional Hamiltonian operator appearing in the isotropic plane magne-toelectroelastic solids is studied. First, all the eigenvaiues and their eigenfunctions in a rectangular domain are solved directly. Then the completeness of the eigenfunction system is proved, which offers a theoretic guarantee of the fea-sibility of variable separation method based on a Hamiltonian system for isotropic plane magnetoelectroelastic solids. Finally, the general solution for the equation in the rectangular domain is obtained by using the symplectic Fourier expansion method.