Simulations of orbital arcs are often performed by making use of classical numerical integration techniques. Although providing highly accurate results for short to medium arc lengths, its validity degrades rapidly with time for long-term calculations. The results, in essence, only yield information on the very special case under consideration. Numerous simulations had to be performed in order to gain insight into the various correlations between force field modeling, orbital configuration, and so on. Analytical orbit integration provides deeper insight into the physical causes of the orbit evolution than any special perturbation technique. It operates directly in the spectral domain rather than in the time domain, and therefore system driving frequencies and corresponding amplitudes and phases are directly detectable. A combination of the numerical and analytical approach can be derived based on Lie series. Here, we focus on the well-known Schwarzschild problem, i.e. a major relativistic effect, and its implications especially for the long-term orbital evolution. For that purpose, an already existing Newtonian approach for the classical two-body problem is being extended to the corresponding relativistic Hamiltonian function. We present a set of expressions which enables a step-wise (in time) computation of the orbit, where the series coefficients itself result from analytical derivations instead of elaborated tables of difference quotients as in case of traditional numerical integrators. Finally, we apply the new algorithm to calculate exemplary orbits, and we also present some performance studies.