  A perfect Euler cuboid is a rectangular parallelepiped with integer edges,
with integer face diagonals, and with integer space diagonal as well. Finding
such parallelepipeds or proving their non-existence is an old unsolved
mathematical problem. Algebraically the problem is described by a system of
Diophantine equations. Symmetry approach to the cuboid problem is based on the
natural $S_3$ symmetry of its Diophantine equations. Factorizing these
equations with respect to their $S_3$ symmetry, one gets some certain ideal
within the ring of multisymmetric polynomials. In the present paper this ideal
is completely calculated and presented through its basis.
