Hamiltonian reduction from particular integrals

We develop a geometric reduction mechanism generated by particular integrals.

A family of functions whose time derivatives close linearly on the same family defines an invariant zero-level submanifold.

In the Hamiltonian case, if this family is in involution, the restricted dynamics is presymplectic, and its characteristic quotient carries a reduced Hamiltonian flow.

This yields a direct bridge between particular integrals, presymplectic reduction, and lower-dimensional Hamiltonian dynamics, and leads to a Liouville-type notion of particular integrability.

We illustrate the framework through mechanical examples and lift constructions, including variants of the Eisenhart lift.