  It is often emphasized that spin-foam models could realize a projection on
the physical Hilbert space of canonical Loop Quantum Gravity (LQG). As a first
test we analyze the one-vertex expansion of a simple Euclidean spin-foam. We
find that for fixed Barbero-Immirzi parameter \gamma=1 the one vertex-amplitude
in the KKL prescription annihilates the Euclidean Hamiltonian constraint of
LQG. Since for \gamma=1 the Lorentzian part of the Hamiltonian constraint does
not contribute this gives rise to new solutions of the Euclidean theory.
Furthermore, we find that the new states only depend on the diagonal matrix
elements of the volume. This seems to be a generic property when applying the
spin-foam projector.
