Machine hammer peening (MHP) is an incremental forming process for surface structuring of technical workpieces or tools. Currently, MHP strongly attracts the attention of automotive and toolmaking industry. Recently performed research showed improved tribological characteristics in lubricated deep drawing in terms of a reduced friction coefficient due to a lubricant pocket effect and a reduced contact area. In order to design the MHP process to obtain an optimized load capacity of the fluid film, the velocity distribution has to be analyzed. Thus, an analytic approach for solving the Reynolds equation as a valid simplification of the Navier-Stokes equations for an elliptic geometry of a MHP structure is proposed in this work. The research assumes stationary hydrodynamic lubrication and is restricted to the longitudinal properties. Thereby, the influence of the structure geometry, the fluid film thickness, the sliding velocity and the dynamic viscosity on the fluid velocity distribution is researched by means of an analytic solution of the Reynolds equation. The derived formula is applied to contribute to the understanding in lubricated deep drawing with MHP structures, but is also transferable on further sliding contacts, e.g. bearing lubrication.