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If $V$ is a vector fiel on $M$, its lift to $T^*M$ is an Hamiltonian vector field with hamiltonian function $V$ (viewed as a linear function on fibres). And if the vector field $V$ is $G$-invariant, it is also invariant as a function on $T^*M$ for the lifted action. This has nothing to do with the fact that $V$ is a gradient.