  In this paper we will solve an open problem raised by Man\'asevich and Mawhin
twenty years ago on the structure of the periodic eigenvalues of the vectorial
$p$-Laplacian. This is an Euler-Lagrangian equation on the plane or in higher
dimensional Euclidean spaces. The main result obtained is that for any exponent
$p$ other than $2$, the vectorial $p$-Laplacian on the plane will admit
infinitely many different sequences of periodic eigenvalues with a given
period. These sequences of eigenvalues are constructed using the notion of
scaling momenta we will introduce. The whole proof is based on the complete
integrability of the equivalent Hamiltonian system, the tricky reduction to
$2$-dimensional dynamical systems, and a number-theoretical distinguishing
between different sequences of eigenvalues. Some numerical simulations to the
new sequences of eigenvalues and eigenfunctions will be given. Several further
conjectures towards to the panorama of the spectral sets will be imposed.
