  We provide a complete quantization for the Gowdy model with local rotational
symmetry in vacuum. We start with a redefinition of the classical constraint
algebra such that the Hamiltonian constraint has a vanishing Poisson bracket
with itself. We apply a canonical quantization within loop quantum gravity and
an improved dynamics scheme. We construct the exact solutions to the
constraints and the physical Hilbert space, together with the physical
observables. The quantization provides a physical picture without
singularities. Besides, a genuine discretization of the spatial geometry
emerges by means of a new quantum observable without classical analogue.
