Petrov type, principal null directions, and Killing tensors of slowly rotating black holes in quadratic gravity

The ability to test general relativity in extreme gravity regimes using gravitational wave observations from current ground-based or future space-based detectors motivates the mathematical study of symmetries black holes modified theories gravity. In this paper we focus on spinning hole solutions two quadratic theories: dynamical Chern-Simons and scalar Gauss-Bonnet We compute principal null directions, Weyl scalars, complex tetrad small-coupling, slow rotation approximation for both theories, confirming that spacetimes are Petrov type I. Additionally, solve Killing equation through rank 6 2 gravity, showing there is no nontrivial tensor those ranks each theory. therefore conjecture still-unknown, exact, quadratic-gravity, black-hole do not possess a fourth constant motion.