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We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model,for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying ameromorphic connection with values in the Langlands dual algebra of the affineLie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $\Psi$-systemamong subdominant solutions. We then use the $\Psi$-system to prove that thegeneralized spectral determinants satisfy the Bethe Ansatz equations of thequantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functionsfor twisted Kac--Moody algebras and we construct new explicit solutions to theBethe Ansatz equations. The paper is a continuation of our previous work on theODE/IM correspondence for simply-laced Lie algebras.

Bethe Ansatz and the Spectral Theory of affine Lie algebra--valued connections II. The non simply--laced case