Invariant Hamiltonians descend to reduced Hamiltonians

$X_H$ is tangent to the level: for every $\xi$, $d{\mu }^{\xi }(X_H)=X_H({\mu }^{\xi })=\{{\mu }^{\xi },H\}=0$ by [F1] and [F2], so $X_H$ lies in $\ker d\mu$ and hence in $T{\mu }^{-1}(\alpha )$ by [F6].

$X_H$ is invariant under the action: since $H$ is $G$-invariant, for $g\in G$ and $p\in M$, ${\omega }_{g\cdot p}(d(a_g{)}_p{X}_H(p),d(a_g{)}_{p}v)={\omega }_p(X_H(p),v)=d{H}_p(v)=d{H}_{g\cdot p}(d(a_g{)}_{p}v)$ for all $v$, so nondegeneracy gives $d(a_g{)}_p{X}_H(p)=X_H(g\cdot p)$.

By steps 1.1 and 1.2 the field $X_H$ is $G_{\alpha }$-invariant and tangent to the level, so $Y_{[p]}:=d{\pi }_p(X_H(p))$ is well defined: for $p^{\mathrm{\prime }}=h\cdot p$ with $h\in G_{\alpha }$ one has $d{\pi }_{p^{\mathrm{\prime }}}{X}_H(p^{\mathrm{\prime }})=d{\pi }_{p^{\mathrm{\prime }}}d(a_h)X_H(p)=d(\pi \circ a_h{)}_p{X}_H(p)=d{\pi }_p{X}_H(p)$ because $\pi \circ a_h=\pi$. It is smooth: by [F4], submersion coordinates for $\pi$ have the form $(u,v)\mapsto u$; fixing $v=v_0$ gives a smooth local section $s$. There $Y=d\pi \circ X_H\circ s$, interpreting $X_H$ as its tangent restriction to the level, so this local expression is smooth.

By invariance, $H{\mathrm{\mid }}_{{\mu }^{-1}(\alpha )}$ is constant on the fibres of $\pi$, so [F4] gives a unique continuous $h:M_{\alpha }\to \mathbb{R}$ with ${\pi }^{\ast }h={\iota }^{\ast }H$. Near every point of $M_{\alpha }$, the submersion $\pi$ has a smooth local section $s$ by [F3] and [F4], by fixing the fibre coordinates as above, and there $h=H\circ \iota \circ s$; hence $h$ is smooth.

The projected field is the Hamiltonian field of $h$: for $v\in T_p{\mu }^{-1}(\alpha )$, $d{h}_{[p]}(d{\pi }_{p}v)=d({\pi }^{\ast }h{)}_p(v)=d({\iota }^{\ast }H{)}_p(v)=d{H}_p(v)={\omega }_p(X_H(p),v)=({\pi }^{\ast }{\omega }_{\alpha }{)}_p(X_H(p),v)={\omega }_{\alpha }(Y_{[p]},d{\pi }_{p}v),$ using [F3]; since $d{\pi }_p$ is onto, ${\iota }_Y{\omega }_{\alpha }=dh$, and uniqueness of Hamiltonian fields [F2] gives $Y=X_h$.

If $\gamma$ is an integral curve of $X_H$ lying in the level, then $\pi \circ \gamma$ is an integral curve of $Y=X_h$ by the chain rule, and by uniqueness of integral curves [F5] it agrees on its interval of definition with the reduced integral curve through the projected initial point. Thus the restricted flow projects wherever the original curve is defined; no completeness or equality of maximal time intervals is asserted. If the level is empty, there is a unique empty descended function and vector field and every assertion is vacuous.