  Conserved quantities are obtained and analyzed in the new models with global
scale invariance recently proposed. Such models allow for non tivial scalar
field potentials and masses for particles, so that the scale symmetry must be
broken somehow. We get to this conclusion by showing that the infrared behavior
of the conserved currents is singular so that there are no conserved charges
associated with the global scale symmetry. The scale symmetry plays
nevertheless a crucial role in determining the structure of the theory and it
implies that in some high field regions the potentials become flat.
