From the Riemann surface of TASEP to ASEP

We consider the asymmetric simple exclusion process (ASEP) with forward hopping rate 1, backward q and periodic boundary conditions. show that Bethe equations of ASEP can be decoupled, at all order in perturbation variable q, by introducing a formal Laurent series mapping roots totally case q=0 (TASEP) to ASEP. The probability height for is then written as single contour integral on Riemann surface which symmetric functions TASEP live.