Spacetimes with prescribed Killing tensor symmetries

In this paper, we investigate spacetime characterized by a hidden symmetry defined by a given Killing tensor. To exhibit this hidden symmetry, the inverse metric must commute with the Killing tensor under the Schouten-Nijenhuis bracket, which translates into a system of partial differential equations (PDEs) for the inverse metric. For some significant examples, we solve these PDEs directly, deriving spacetimes with prescribed Killing tensor symmetries, including those specified by higher-rank Killing tensors. Utilizing these symmetries, we study related problems such as null geodesics, photon region, and separation of variables of wave equations. Through this work, we aim to demonstrate that hidden symmetry is more accessible than previously believed.