One day workshop in honor of Maxim Pavlov's 60th birthday. Growth of Lie algebras and integrability

We consider naturally graded Lie algebras ${\displaystyle {\mathfrak {g}}=\oplus _{i=1}^{n}{\mathfrak {g}}_{i},\;[{\mathfrak {g}}_{1},{\mathfrak {g}}_{i}]={\mathfrak {g}}_{i+1},\;i\geq 1.}$
In the finite-dimensional case they are called Carnot algebras and play an important role in non-holonomic geometry and geometric control theory. A naturally graded Lie algebra ${\displaystyle {\mathfrak {g}}}$ is generated by ${\displaystyle {\mathfrak {g}}_{1}}$ and one can define its natural growth function ${\displaystyle F_{\mathfrak {g}}^{gr}(n)=\sum _{i=1}^{n}\dim {{\mathfrak {g}}_{i}}}$ which is well-defined.
It turned out that the characteristic Lie algebras ${\displaystyle \chi }$ of some nonlinear hyperbolic partial differential equations are precisely such positively graded Lie algebras. The integability of these equations in the sense of Darboux or higher symmetries leads to the slow growth of ${\displaystyle \chi }$.
I will also try to discuss another geometric integrability, the integrability of complex structures on Carnot algebras. It turns out that in this case, on the contrary, Lie algebras must grow sufficiently fast.