My key recent publication
New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376 (arXiv:1401.2122)
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.
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; for further details please see the paper
Integrable (3+1)-dimensional system with an algebraic Lax pair, Appl. Math. Lett. 92 (2019), 196-200. (arXiv:1812.02263)
You may wish to look at the recent slides for additional background and motivation before proceeding to the above articles themselves.