Evolutionary differential equations and symmetries of systems of finite type

The talk is devoted to systems of evolutionary differential equations of the form $u_t=f(x, u_x, u_{xx}, \dots)$. Here $x=(x_1,\dots, x_n)$ and $u=(u_1,\dots, u_m)$. These systems define flows on the maximal integral manifolds of some completely integrable distributions.
For example, if $n=m=1$, then the evolution equation defines the symmetries of the ordinary differential equation $F(x,y,y',\dots, y^{(k)})=0$. This allows, knowing the solution of the ODE, to construct classes of exact or approximate solutions of the evolution equation even in those cases when it does not have the necessary margin of symmetries. The first results in this direction were obtained by B. Kruglikov, V. Lychagin and O. Lychagina back in the 2000s.
In the case when $n>1$, systems of finite type should be used instead of ODEs. Examples from physics and biology will be given.