ODE/IM correspondence for modified $B_2^{\left(1\right)}$ affine Toda field equation

We study the massive ODE/IM correspondence for modified $B_2^{\left(1\right)}$ affine Toda field equation. Based on the ψ-system for the solutions of the associated linear problem, we obtain the Bethe ansatz equations. We also discuss the T–Q relations, the T-system and the Y-system, which are shown to be related to those of the $A_3/Z_2$ integrable system. We consider the case that the solution of the linear problem has a monodromy around the origin, which imposes nontrivial boundary conditions for the T-/Y-system. The high-temperature limit of the T- and Y-system and their monodromy dependence are studied numerically.