We study Bailey pairs construction for hyperbolic hypergeometric integral
identities acquired via the duality of lens partitions functions for the
three-dimensional $\mathcal N=2$ supersymmetric gauge theories on
$S_b^3/\mathbb{Z}_r$. The novel Bailey pairs are constructed for the
star-triangle relation, the star-star relation and the pentagon identity. The
first two of them are integrability conditions for the Ising-type integrable
lattice models. The last one corresponds to the representation of the basic
$2-3$ Pachner move for triangulated 3-manifolds.
