The problem of the evolution of an electromagnetic TEM wave (transverse wave) in a nonlinear anisotropic medium (between infinite conducting planes or a coaxial waveguide) is solved. It is believed that an electromagnetic wave is excited due to a boundary condition on a flat boundary perpendicular to the direction of propagation of the wave (for example, along the axis of the waveguide). To describe the behavior of a TEM wave in a nonlinear medium, a fairly simple model is used when the dependence of the electric induction D on the electric field strength E includes quadratic terms (polarization P ∼ E2), which is typical for an anisotropic medium (in a nonlinear isotropic medium, such a dependence can include only terms with odd degrees E). The relation of the magnetic induction B with the magnetic field strength H is assumed to be linear for simplicity. It is shown that in the process of wave propagation, the occurrence of shock electromagnetic waves occurs. The moment of time and the coordinate for which the formation of a shock wave is possible are indicated. From a mathematical point of view, the process is described by a system of two spatially one-dimensional quasi-linear hyperbolic partial differential equations of the first order, for which boundary conditions are set. The implicit and explicit solution of the problem is constructed using the hodograph method based on the conservation law. The system of quasi-linear equations is transformed into one linear differential equation in partial derivatives of the second order with variable coefficients. The Riemann invariants and the Riemann-Green function are specified for a linear equation in which the independent variables are the Riemann invariants, and the unknown functions are time and coordinate. In this case, implicit solutions are obtained in analytical form, and the explicit solution is constructed on the level lines of the implicit solution, by numerical integration of some Cauchy problem for a system of ordinary differential equations. We emphasize that no approximations typical for finite-difference methods, finite element methods and finite volumes are used and the accuracy of the numerical solution is limited only by the accuracy of methods for solving ordinary differential equations. For boundary conditions corresponding to the fading time pulse the results of calculations describing the evolution of TEM waves are presented, and the time points (and coordinates) at which the occurrence of shock electromagnetic waves is possible are indicated.