  We study the two-dimensional affine Toda field equations for affine Lie
algebra $\hat{\mathfrak{g}}$ modified by a conformal transformation and the
associated linear equations. In the conformal limit, the associated linear
problem reduces to a (pseudo-)differential equation. For classical affine Lie
algebra $\hat{\mathfrak{g}}$, we obtain a (pseudo-)differential equation
corresponding to the Bethe equations for the Langlands dual of the Lie algebra
$\mathfrak{g}$, which were found by Dorey et al. in study of the ODE/IM
correspondence.
