The Maxwell - Lorenz system of an electromagnetic field interacting with charged particles (point charges)
is considered in the Darwin approximation which is characterized by the Lagrangian and Hamiltonian of
the particles both uncoupled with the field. The solution of the equation of motion of the particles with the approximated Darwin Hamiltonian is found on a finite time interval with the use of the Cauchy theorem.
Components of this solution are represented as holomorphic functions of time.