The first direct electromagnetic observation of a black hole (BH) was done by the EHT collaboration in 2019, triggering the possibility to test General Relativity (GR) in its strong field regime by probing the curvature of spacetime close to BHs. The observed image contains in particular the "critical curve", which is the imprint of the photons emitted by the accretion disk and trapped around the BH for several orbits. Measuring this critical curve and comparing it to the GR prediction constitutes a new kind of test of this theory: it is therefore crucial to compute this observable for beyond-GR BHs. In this work, we compute analytically such a critical curve for the "Kerr off-shell" family of spacetimes which describes a broad class of rotating BHs beyond the Kerr solution. This family of solutions is the most general extension of Kerr that preserves its "Killing tower" of symmetries, making the geodesic motion integrable. We study several concrete examples, studying in particular for the first time solutions with polar deformations with respect to Kerr. We conclude on the feasibility of a test of GR using future critical curve observations.
