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This paper extends the correspondence between discrete Cluster IntegrableSystems and BPS spectra of five-dimensional $\mathcal{N}=1$ QFTs on$\mathbb{R}^4\times S^1$ by proving that algebraic solutions of the integrablesystems are exact solutions for the system of TBA equations arising from theBPS spectral problem. This statement is exemplified in the case of M-theorycompactifications on local del Pezzo Calabi-Yau threefolds, corresponding toq-Painlev\'e equations and $SU(2)$ gauge theories with matter. A degenerationscheme is introduced, allowing to obtain closed-form expression for the BPSspectrum also in systems without algebraic solutions. By studying the exampleof local del Pezzo 3, it is shown that when the region in moduli spaceassociated to an algebraic solution is a ``wall of marginal stability'', theBPS spectrum contains states of arbitrarily high spin, and corresponds to a 5duplift of a four-dimensional nonlagrangian theory.

BPS Spectra and Algebraic Solutions of Discrete Integrable Systems