Tau--Function Multilinear Hierarchy of the Tomimatsu--Sato Spacetime: A Gravitational Realization of the YTSF Integrable Structure

Abstract

The Tomimatsu--Sato (TS) family generalizes the Kerr black hole to higher multipole order $\delta$ and has long been regarded as algebraically complicated without any clear integrability. We show instead that stationary axisymmetric vacuum Einstein equations, when the Ernst potential is written as a $\tau$--ratio $\mathcal{E}=\tau_1/\tau_0$, admit a universal decomposition of the Ernst numerator into a cubic part containing all second derivatives and a quartic \emph{gradient envelope}. The cubic sector can be written in terms of $Z_3$--symmetric trilinear Hirota operators, revealing a hidden integrable structure. For $\delta=2$, using the explicit Tomimatsu--Sato polynomials, we verify that this trilinear sector coincides with a Yu--Toda--Sasa--Fukuyama (YTSF) equation-type kernel. Thus the TS geometry forms a gravitational realization of a multilinear $\tau$--function hierarchy in stationary axisymmetric general relativity.