A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. We study a system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas, a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid, a system of Maxwell equations for vacuum, and a system of gas dynamics equations in cylindrical coordinates. We show how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).