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The basic equation to describe transport of cosmic rays in a random magnetic field parallel to a magnetic background field is the twodimensional Fokker-Planck equation. In this paper an investigation of this equation is presented to improve understanding of particle transport, to depict the process of parallel diffusion and to improve recently derived nonlinear transport theories. Article published by EDP Sciences and available at http://www.edpsciences.org/aa or http://dx.doi:10.1051/0004-6361:20053664 fatcat:rhc54viakvhftkheqzco44ias4