  We study the root of unity limit of the lens elliptic gamma function solution
of the star-triangle relation, for an integrable model with continuous and
discrete spin variables. This limit involves taking an elliptic nome to a
primitive $rN$-th root of unity, where $r$ is an existing integer parameter of
the lens elliptic gamma function, and $N$ is an additional integer parameter.
This is a singular limit of the star-triangle relation, and at subleading order
of an asymptotic expansion, another star-triangle relation is obtained for a
model with discrete spin variables in $\mathbb{Z}_{rN}$. Some special choices
of solutions of equation of motion are shown to result in well-known discrete
spin solutions of the star-triangle relation. The saddle point equations
themselves are identified with three-leg forms of "3D-consistent" classical
discrete integrable equations, known as $Q4$ and $Q3_{(\delta=0)}$. We also
comment on the implications for supersymmetric gauge theories, and in
particular comment on a close parallel with the works of Nekrasov and
Shatashvili.
