A self-similar solution for the two-dimensional Broadwell system via the Bateman equation

A self-similar solution of the Broadwell system is found. Here the solution is sought using a reduction that transforms the given system into a system of differential equations. Further, the solution is constructed using the Painlev\'e series. Here the system already passes the Painlev\'e test and it is possible to find the solution if the equations in resonance satisfy the solution of the two-dimensional Bateman equation. Exact solution of the Bateman equation is established, allowing to find new explicit solution for the original system. In the process of calculations, we use the Wolfram Mathematica program. The proof of these results is carried out at a rigorous mathematical level.