However, the overwhelming majority of new results and theoretical understanding pertain only to PDE systems with two independent variables. The situation for PDE systems with more than two independent variables turns out to be much more complicated due to gauge freedom relating potential variables.
We present a systematic treatment of nonlocally related PDE systems with $n\geq 3$ independent variables, and compute new examples of nonlocal symmetries, nonlocal conservation laws, and exact solutions for such systems.
This is joint work with George Bluman (UBC).
The symmetry investigation of the Kontsevich system that was performed includes the determination
of Lax Pairs of which 10 were found, each providing an infinite sequence of
conservation laws and of symmetries. It also includes the direct computation
of all inhomogeneous symmetries up to degree 15. The corresponding linear
algebraic systems that have been solved contain up to 300 million equations
for 59 million undetermined coefficients.