In this work we aim to investigate non-mainstream thick tori configurations around Kerr Black Holes with Scalar Hair (KBHsSH). For that goal, we provide a first approach using constant specific angular momentum non-self-gravitating Polish doughnuts. Through a series of examples, we show the feasibility of new topologies, such as double-centered tori with two cusps as well as similar structures as the ones found for rotating Boson Stars (BSs), namely tori endowed with two centers and a single cusp. These KBHsSH solutions are also shown to possibly house static surfaces, associated to the static rings present in these spacetimes. Through this report we highlight the differences between these fluid configurations when housed by some KBHsSH examples, standard Kerr black holes and rotating BSs.
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