The recent article
Multidimensional integrable systems from contact geometry
addresses in a positive fashion the longstanding problem of search for partial differential systems in four independent variables ((3+1)D or 4D for short) that are integrable in the sense of soliton theory. This problem is highly important as according to Einstein's general relativity our spacetime is four-dimensional and thus the (3+1)D case is especially relevant for applications.Â
In the above article it is proved that integrable (3+1)D systems are significantly less exceptional than it appeared before: in addition to a handful of previously known important yet isolated examples like the (anti)self-dual Yang--Mills equations there is a large new class of integrable (3+1)D systems with Lax pairs of a novel kind related to contact geometry, and explicit form of two infinite families of integrable (3+1)D systems from this class with polynomial and rational Lax pairs is given.
Moreover, the said new class contains inter alia the first known example of an integrable (3+1)D system with a nonisospectral Lax pair which is algebraic, rather than just rational, in the variable spectral parameter.
You may wish to look at the recent slides for additional background and motivation before proceeding to the above article itself.