  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 $A_2$-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 $\phi_{12}$, $\phi_{21}$
and $\phi_{15}$. This is checked against previous results obtained using the
thermodynamic Bethe ansatz.
