Symplectic mechanics of relativistic spinning compact bodies. I. linear-in-spin integrability under Killing-Yano symmetry

We study the Hamiltonian dynamics of a neutral, massive spinning test body at linear order in spin, as governed by the Mathisson--Papapetrou--Tulczyjew--Dixon equations in a 4-dimensional spacetime admitting a non-degenerate Killing--Yano tensor.

The Tulczyjew--Dixon spin supplementary condition is imposed as a set of algebraic constraints, and the resulting constraint surface is equipped with a Poisson--Dirac bracket, yielding a non-degenerate, 10-dimensional physical phase space.

On this reduced space we identify five functionally independent first integrals in involution: the autonomous Hamiltonian, two constants of motion associated with two commuting Killing vectors, a generalised Carter constant, and the Rüdiger constant; all four descending from the Killing--Yano tensor.

All calculations are covariant, and the result is a purely geometric statement: it requires no background field equations and holds for any metric admitting a non-degenerate Killing--Yano tensor, beyond Kerr, beyond vacuum and beyond general relativity.

Our results identify Killing--Yano symmetry as the sole geometric source of linear-in-spin integrability.

Extensions to quadratic-in-spin dynamics, including the spin-induced quadrupole, and to tidally-induced quadrupolar effects for non-spinning bodies, are treated in companion papers.