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