- Abstract Plasma filamentation is often encountered in collisionless shocks and inertial confinement fusion. We develop a general analytical description of the two-dimensional relativistic filamentary equilibrium and derive the conditions for existence of potential-free equilibria. A pseudopotential equation for the vector-potential is constructed for cold and relativistic Maxwellian distributions. The role of counter-streaming is explained. We present single current sheet and periodic current sheet solutions, and analyze the equilibria with electric potential. These solutions can be used to study linear and nonlinear evolution of the relativistic filamentation instability.