The Zel'dovich approximation is very efficient in reconstructing the initial conditions (in the form of a displacement field, a deformation tensor, or the corresponding potential) from observational data, usually proper velocities. Recently it has been shown that this problem reduces to a differential equation for the velocity potential, the Bernouilli-Zel'dovich equation, and numerical methods were proposed to solve it. Here I derive an analytical solution to this equation which allows reconstruction without any numerical integration (except a spatial integration of the present velocity field which is the first step of any procedure). This allows the reconstruction of the initial potential at Lagrangian positions