Exact Solutions of two integrable Cellular Automata with localized impurity

We study a 1+1 dimensional integrable reversible cellular automaton (RCA54) with an impurity cell governed by a different local rule. Building upon prior works, using matrix product and patch state ansatz techniques, we derive exact analytical solutions for the non-equilibrium steady states and identify eigenstates associated with -1 eigenvalues, which correspond to discrete time-crystal phases. Coupling the system to stochastic boundary baths yields a Markov chain with deterministic bulk and stochastic boundary dynamics, where boundary parameters uniquely determine steady and time-crystal states. Our results provide exact insights into impurity dynamics in integrable discrete systems, revealing degenerate steady-state manifolds and emergent complex behavior from simple local interactions. This work highlights the potential of integrable RCAs as discrete models for exploring non-equilibrium and impurity phenomena in mathematical physics and opens promising directions for future theoretical studies of impurity problems using discrete integrable models.