Reduced phase space approach to the U(1)3 model for Euclidean quantum gravity

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