The article provides a very clear introduction to the Lie group analysis of differential equations. Rather instructive examples are thoroughly discussed. The Navier-Stokes-Fourier equations for a viscous and heat conducting monatomic gas in a rotating frame are reduced to two different autonomous forms, and some explicit solutions are given. The Monge-Ampère equation with constant coefficients is reduced to a linear form.
The concepts of Lie remarkable PDE and of completely exceptional PDE are introduced, however, we can state only the following indications. The second order equation \[ \bigtriangleup (x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy})=0 \] admits the 15–parameter Lie algebra of symmetries \[ \partial/\partial a,a\partial/\partial a,a\partial/\partial b,a(x\partial/\partial x+y\partial/\partial y+u\partial/\partial u)\quad (a,b=x,y,u) \] if and only if it is equivalent to the Monge-Ampère equation \(u_{xx}u_{yy}-u^2_{xy}=0,\) so the latter equation is called Lie exceptional. The completely exceptional nonlinear hyperbolic equations are defined in terms of the behaviour of discontinuities of solutions but they can be also characterized in terms of the Lie point symmetries.