The search for partial differential systems in four independent variables that are integrable in the sense of soliton theory is a longstanding problem of mathematical physics, naturally motivated by the fact that our spacetime is four-dimensional according to Einstein's general relativity; such systems are referred to as (3+1)-dimensional integrable systems or 4D integrable systems or, in terms used in physics, 4D classically integrable field theories, in general nonrelativistic and nonlagrangian.
The recent publication Multidimensional integrable systems from contact geometry gives a comprehensive review of our results addressing this problem in a positive fashion and shows, in particular, that integrable (3+1)-dimensional systems are significantly less exceptional than it appeared before: it turns out that in addition to a handful of well-known important yet isolated examples like the (anti-)self-dual Yang--Mills equations or (anti-)self-dual vacuum Einstein equations there is a large new class containing infinitely many integrable (3+1)-dimensional systems with Lax pairs of a novel kind related to contact geometry.
You may wish to look at these slides for additional background and motivation before proceeding to the article itself.
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