  We examine some excited state energies in the non-unitary integrable quantum
field theory obtained from the perturbation of the minimal conformal field
theory model $M_{3,5}$ by its operator $\phi_{2,1}$. Using the correspondence
of this IQFT to the scaling limit of the dilute $A_2$ lattice model (in a
particular regime) we derive the functional equations for the QFT commuting
transfer matrices. These functional equations can be transformed to a closed
set of TBA-like integral equations which determine the excited state energies
in the finite-size system. In particular, we explicitly construct these
equations for the ground state and two lowest excited states. Numerical results
for the associated energy gaps are compared with those obtained by the
truncated conformal space approach (TCSA).
