  We discuss a general procedure to construct an integrable real-time
trotterization of interacting lattice models. As an illustrative example we
consider a spin-$1/2$ chain, with continuous time dynamics described by the
isotropic ($XXX$) Heisenberg Hamiltonian. For periodic boundary conditions
local conservation laws are derived from an inhomogeneous transfer matrix and a
boost operator is constructed. In the continuous time limit these local charges
reduce to the known integrals of motion of the Heisenberg chain. In a simple
Kraus representation we also examine the nonequilibrium setting, where our
integrable cellular automaton is driven by stochastic processes at the
boundaries. We show explicitly, how an exact nonequilibrium steady state
density matrix can be written in terms of a staggered matrix product ansatz.
This simple trotterization scheme, in particular in the open system framework,
could prove to be a useful tool for experimental simulations of the lattice
models in terms of trapped ion and atom optics setups.
