The reduction you require *is* a (very) special case of Marsden-Weinstein ([1974](http://www.cds.caltech.edu/~marsden/bib/1974/01-MaWe1974/)). Your one constant of the motion, say $\psi$, is the moment map of an action of the additive group $G=\mathbf R$ on the symplectic manifold with coordinates say $(x_1,y_1,\dots,x_n,y_n)$ — viz. $\psi$’s hamiltonian flow, obtained by solving
$$
\frac{d}{dt}\begin{pmatrix}x_i\\y_i\end{pmatrix}=
\begin{pmatrix}-\partial\psi/\partial y_i\\\phantom{-}\partial\psi/\partial x_i\end{pmatrix}.
\tag1
$$
Their Theorem 1 says that each level $\psi^{-1}(\mu)$ is a coisotropic submanifold whose null leaves are the $G$-orbits, so the symplectic form descends to the leaf space $\psi^{-1}(\mu)\,/\,G$. This is the reduced space, of dimension 2n – 2. Their Theorem 2 adds that $H$, being constant on leaves since $\{H,\psi\}=0$, descends to a function $H_\mu$ on the reduced space. This is the reduced system.

(This is all subject to technical conditions: (1) complete, $\mu$ “weakly regular” value of $\psi$, $G$-action on levels free and proper — which I don’t think matter much in your coordinate formulation. E.g. taking $\psi$ as $p_n$, the subquotient means “fix $p_n$ and ignore $q_n$”, and Darboux charts on the symplectic manifold $\psi^{-1}(\mu)\,/\,G$ give your desired new coordinates $(p_1,q_1,\dots,p_{n-1},q_{n-1})$.)

As their introduction points out, this special case $G=\mathbf R$ had long been known as the theory of “ignorable coordinates”, exposed with plenty of examples in e.g. Whittaker ([1904](//zbmath.org/?q=an:35.0682.01), [§38 *sq*](//archive.org/details/treatiseonanalyt00whit_0/page/53)). Other nice example from Souriau, who had the theory for abelian $G$ ([1970](//ams.org/mathscinet-getitem?mr=41:4866), [Chap. III](http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm), 12.153 *sq*): at a negative level of a hydrogen atom’s energy $\psi$, the reduced space is $\smash{\mathrm S^2\times\mathrm S^2}$, and any component $H$ of angular momentum or the eccentricity (a.k.a. Lenz) vector descends there (together they generate an $\mathrm{SO}(4)$ action on the subquotient).