  The developments in this paper are concerned with nonholonomic field theories
in the presence of symmetries. Having previously treated the case of vertical
symmetries, we now deal with the case where the symmetry action can also have a
horizontal component. As a first step in this direction, we derive a new and
convenient form of the field equations of a nonholonomic field theory.
Nonholonomic symmetries are then introduced as symmetry generators whose
virtual work is zero along the constraint submanifold, and we show that for
every such symmetry, there exists a so-called momentum equation, describing the
evolution of the associated component of the momentum map. Keeping up with the
underlying geometric philosophy, a small modification of the derivation of the
momentum lemma allows us to treat also generalized nonholonomic symmetries,
which are vector fields along a projection. Such symmetries arise for example
in practical examples of nonholonomic field theories such as the Cosserat rod,
for which we recover both energy conservation (a previously known result), as
well as a modified conservation law associated with spatial translations.
