A biquadratic Diophantine equation associated with perfect cuboids

A perfect Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. Such cuboids are not yet discovered and their non-existence is also not proved. Perfect Euler cuboids are described by a system of four Diophantine equation possessing a natural $S_3$ symmetry. Recently these equations were factorized with respect to this $S_3$ symmetry and the factor equations were derived. In the present paper the factor equations are transformed to $E$-form and then reduced to a single biquadratic equation.