  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 hidden symmetries, including
those specified by higher-rank Killing tensors. Utilizing the hidden
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.
