Killing tensors of Weyl's class

Abstract

This work represents an initial step towards a systematic framework for identifying hidden symmetries in algebraically general (Petrov type I) spacetimes. Motivated by this, we focus on Weyl's class, which provides a simple yet a class of algebraically general solutions of static, axisymmetric geometries. Employing the canonical forms of rank-2 Killing tensors, we systematically derive the constraints that must be satisfied for these spacetimes to admit genuine hidden symmetries. Our analysis yields an analytical characterization of the obstructions to integrability within subclasses of Weyl's class, both in vacuum and electrovacuum. In particular, we find that no two-Killing-vector member of Weyl's class, in vacuum or electrovacuum, admits an irreducible Killing tensor capable of supplying the additional integral of motion required for complete integrability. Members of Weyl's class admitting a third additional Killing-vector isometry (i.e. Levi-Civita) are already completely integrable through their manifest symmetries alone admitting only reducible Killing tensors. This result provides an analytic complement to previous analytical and numerical investigations, and establishes a systematic connection between the absence of hidden symmetries and the breakdown of complete integrability. Beyond the specific classification obtained here, these results demonstrate the potential of our approach regarding the canonical forms of Killing tensor as a systematic tool for probing hidden symmetries in Petrov type I spacetimes.