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We study rapid-turn trajectories in a class of two-field cosmological models,whose scalar manifold is the Poincar\'e disk. Background solutions inmulti-field inflation, with field-space trajectories exhibiting sharp turns,can seed primordial black hole (PBH) formation. We investigate a class of exactsolutions with hidden symmetry and show that they exhibit the kind of transientrapid-turn period, needed to induce PBH generation. Furthermore, we relax thesymmetry condition and find, in a certain regime, modified solutions withimproved behavior of the Hubble $\eta$-parameter, which preserve the desiredshape of the turning rate function. Interestingly, the modified solutionsdescribe a brief ultra-slow roll phase, followed by long-term slow rollinflation. It is notable that slow roll occurs near the center (not near theboundary) of the Poincar\'e disk, unlike in the standard $\alpha$-attractorcase.

On Primordial Black Holes from Rapid Turns in Two-field Models