  The present paper introduces the analysis of the eigenvalue problem for the
elasticity equations when the so called Navier-Lam\'e system is considered.
Such a system introduces the displacement, rotation and pressure of some linear
and elastic structure. The analysis of the spectral problem is based in the
compact operators theory. A finite element method based in polynomials of
degree $k\geq 1$ are considered in order to approximate the eigenfrequencies
and eigenfunctions of the system. Convergence and error estimate are presented.
An a posteriori error analysis is performed, where the reliability and
efficiency of the proposed estimator is proved. We end this contribution
reporting a series of numerical tests in order to assess the performance of the
proposed numerical method, for the a priori and a posteriori estimates.
