  Exact solutions of interacting random walk models, such as 1D lattice gases,
offer precise insight into the origin of nonequilibrium phenomena. Here, we
study a model of run-and-tumble particles on a ring lattice interacting via
hardcore exclusion. We present the exact solution for one and two particles
using a generating function technique. For two particles, the eigenvectors and
eigenvalues are explicitly expressed using two parameters reminiscent of Bethe
roots, whose numerical values are determined by polynomial equations which we
derive. The spectrum depends in a complicated way on the ratio of direction
reversal rate to lattice jump rate, $\omega$. For both one and two particles,
the spectrum consists of separate real bands for large $\omega$, which mix and
become complex-valued for small $\omega$. At exceptional values of $\omega$,
two or more eigenvalues coalesce such that the Markov matrix is
non-diagonalizable. A consequence of this intricate parameter dependence is the
appearance of dynamical transitions: non-analytic minima in the longest
relaxation times as functions of $\omega$ (for a given lattice size).
Exceptional points are theoretically and experimentally relevant in, e.g., open
quantum systems and multichannel scattering. We propose that the phenomenon
should be a ubiquitous feature of classical nonequilibrium models as well, and
of relevance to physical observables in this context.
