  If one replaces the constraints of the Ashtekar-Barbero $SU(2)$ gauge theory
formulation of Euclidean gravity by their $U(1)^3$ version, one arrives at a
consistent model which captures significant structure of its $SU(2)$ version.
In particular, it displays a non trivial realisation of the hypersurface
deformation algebra which makes it an interesting testing ground for
(Euclidean) quantum gravity as has been emphasised in a recent series of papers
due to Varadarajan et al. In this paper we consider a reduced phase space
approach to this model. This is especially attractive because, after a
canonical transformation, the constraints are at most {\it linear} in the
momenta. In suitable gauges, it is therefore possible to find a closed and
explicit formula for the physical Hamiltonian which depends only on the
physical observables. Not surprisingly, that physical Hamiltonian is
generically neither polynomial nor spatially local. The corresponding reduced
phase space quantisation can be confronted with the constraint quantisation due
to Varadarajan et al to gain further insights into the quantum realisation of
the hypersurface deformation algebra.
