Algorithm for finding explicit solutions of overdetermined systems of differential equations

Previously, the authors proposed a general method for finding particular solutions for overdetermined systems of partial differential equations (PDE), where the number of equations is greater than the number of unknown functions. In this paper, an algorithm for finding solutions for overdetermined PDE systems is proposed, where the authors use a method for finding an explicit solution for overdetermined algebraic (polynomial) equations. Using this algorithm, some overdetermined PDE systems can be solved in explicit form. The main difficulty of this algorithm is the huge number of polynomial equations that arise, which need to be investigated and solved numerically or explicitly. For example, the overdetermined hydrodynamic equations obtained earlier by the authors give at least 10 million such equations. However, if the equations are solved explicitly, then it is possible to write out the solution of the hydrodynamic equations in a general form, which is of great scientific interest.