  We apply the algebraic quantization programme proposed by Ashtekar to the
analysis of the Belinski\v{\i}-Zakharov classical spacetimes, obtained from the
Kasner metrics by means of a generalized soliton transformation. When the
solitonic parameters associated with this transformation are frozen, the
resulting Belinski\v{\i}-Zakharov metrics provide the set of classical
solutions to a gravitational minisuperspace model whose Einstein equations
reduce to the dynamical equations generated by a homogeneous Hamiltonian
constraint and to a couple of second-class constraints. The reduced phase space
of such a model has the symplectic structure of the cotangent bundle over
$I\!\!\!\,R^+\times I\!\!\!\,R^+$. In this reduced phase space, we find a
complete set of real observables which form a Lie algebra under Poisson
brackets. The quantization of the gravitational model is then carried out by
constructing an irreducible unitary representation of that algebra of
observables. Finally, we show that the quantum theory obtained in this way is
unitarily equivalent to that which describes the quantum dynamics of the Kasner
model.
