  A rational perfect cuboid is a rectangular parallelepiped whose edges and
face diagonals are given by rational numbers and whose space diagonal is equal
to unity. Its existence is equivalent to the existence of a perfect cuboid with
all integer edges and diagonals. Finding such a cuboid or proving its
non-existence is an old unsolved problem. Recently, based on a symmetry
approach, the equations of a perfect cuboid were transformed to factor
equations. The factor equations turned out to be solvable and, being solved,
have led to a pair of inverse problems. Our efforts in the present paper are
toward solving these inverse problems. Algebraic parametrizations for their
solutions using algebraic functions of two rational arguments are found.
