Conformal Killing--Yano Ricci solitons: Structure, compatibility, and rigidity
We introduce a geometric structure -- a conformal Killing--Yano Ricci soliton (CKY--RS) -- that couples conformal Ricci soliton (CRS) geometry to conformal Killing--Yano (CKY) 2--forms. The soliton field of the CRS geometry is given by the divergence of the CKY 2--form. We introduce a conserved CKY--Cotton current and derive a compatibility identity relating the Cotton tensor, the CRS obstruction tensor, and the CKY 2--form. In 4--dimensional Lorentzian signature, we show that, under non-degeneracy and closedness assumptions on the CKY form, a CKY--RS structure forces the conformal representative to be locally Kerr--NUT--(A)dS. For a closed non-degenerate CKY on a Kerr--NUT--(A)dS background, the conformal deformation is necessarily trivial. For Einstein backgrounds of arbitrary dimension and signature, the conformal factor satisfies an eigenvalue equation and an Obata--type Hessian equation. If the background is also compact or a CKY orbit is periodic, the conformal factor is an invariant of the CKY--flow and we obtain simple spectral obstructions to non-trivial CKY--RS structures. From the Hessian equation, we obtain obstruction and classification results for the non-trivial conformal sector, including product/Brinkmann geometries and a Weyl--aligned branch. Finally, we give explicit constructions for static spherically symmetric geometries and BTZ backgrounds, including a CKY--RS realization with a time-dependent conformally flat representative. These results provide a geometric framework for studying CRS with hidden symmetry structure, with potential applications to exact geometries in general relativity.