Differential equations and integrable models: the SU(3) case

We exhibit a relationship between the massless a(2)2 integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schr\"odinger equation. This forms part of a more general correspondence involving A2-related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the nonlinear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators ϕ12, ϕ21 and ϕ15. This is checked against previous results obtained using the thermodynamic Bethe ansatz.