Groundwater flow with steep gradients in a vertical plane of infinite horizontal extension due to arbitrary non-symmetric strip sources and/or sinks can be described by the 2D Laplace equation. Notwithstanding the strongly nonlinear character of the free surface boundary condition, the exact analytical solution to this problem is developed in a closed form by employing neither the Dupuit assumption nor any other form of linearization. The first section of the development, still including the unsteady case, leads via conformal mapping and transformation procedures to a singular integro-differential-equation for the transient groundwater table. From this point onwards we restrict ourselves to the steady case for which the exact solution of the 2D Laplace equation for the pressure head and the location of the groundwater table was achieved. The solution is expressed exclusively in algebraic terms without the need for iterative procedures. It can not only be applied to real world phenomena, including a simple solution of the inverse problem, but also provide a new transparency regarding the solution characteristics and may serve as a standard for investigating numerical solutions and the domain of validity of simplified approaches. The computer program can be downloaded from www.tu-dresden.de/fghhihm/hydrologie.html.