  We consider a unitary circuit where the underlying gates are chosen to be
R-matrices satisfying the Yang-Baxter equation and correlation functions can be
expressed through a transfer matrix formalism. These transfer matrices are no
longer Hermitian and differ from the ones guaranteeing local conservation laws,
but remain mutually commuting at different values of the spectral parameter
defining the circuit. Exact eigenstates can still be constructed as a Bethe
ansatz, but while these transfer matrices are diagonalizable in the
inhomogeneous case, the homogeneous limit corresponds to an exceptional point
where multiple eigenstates coalesce and Jordan blocks appear. Remarkably, the
complete set of (generalized) eigenstates is only obtained when taking into
account a combinatorial number of nontrivial vacuum states. In all cases, the
Bethe equations reduce to those of the integrable spin-1 chain and exhibit a
global SU(2) symmetry, significantly reducing the total number of eigenstates
required in the calculation of correlation functions. A similar construction is
shown to hold for the calculation of out-of-time-order correlations.
