  A perfect cuboid is a rectangular parallelepiped with integer edges and
integer face diagonals whose space diagonal is also integer. The existence of
such cuboids is neither proved, nor disproved. A rational perfect cuboid is a
natural companion of a perfect cuboid absolutely equivalent to the latter one.
Its edges and face diagonals are rational numbers, while its space diagonal is
equal to unity. Recently, based on a symmetry reduction, it was shown that
edges of a rational perfect cuboid are roots of a certain cubic equation with
rational coefficients depending on two rational parameters. Face diagonals of
this cuboid are roots of another cubic equation whose coefficients are rational
numbers depending on the same two rational parameters. In the present paper
these two cubic equations are studied for reducibility. Six special cases of
their reducibility over the field of rational numbers are found.
