  We study the existence of first integrals in nonholonomic systems with
symmetry. First we define the concept of $\mathcal{M}$-cotangent lift of a
vector field on a manifold $Q$ in order to unify the works [Balseiro P., Arch.
Ration. Mech. Anal. 214 (2014), 453-501, arXiv:1301.1091], [Fass\`o F., Ramos
A., Sansonetto N., Regul. Chaotic Dyn. 12 (2007), 579-588], and [Fass\`o F.,
Giacobbe A., Sansonetto N., Rep. Math. Phys. 62 (2008), 345-367]. Second, we
study gauge symmetries and gauge momenta, in the cases in which there are the
symmetries that satisfy the so-called vertical symmetry condition. Under such
condition we can predict the number of linearly independent first integrals
(that are gauge momenta). We illustrate the theory with two examples.
