Maciej Dunajski

TL;DR: In this article, it was shown that the Einstein-Weyl equations in 2+1 dimensions contain the dispersionless Kadomtsev-Petviashvili (dKP) equation as a special case: if an EW structure admits a constant-weighted vector, then it is locally given by h = d y 2 −4 d x d t−4u d t −4 u d t 2, where u=u(x,y,t) satisfies the dKP equation (ut−uux)x=uyy.

TL;DR: The integrability of the hyperCR Einstein-Weyl equations in 2+1 dimensions was established in this paper, where a pair of quasi-linear PDEs of hydrodynamic type were constructed from a twistor correspondence.

TL;DR: In this article, the integrability in classical mechanics has been studied in the context of conformal structures and asymmetric reductions in the Lagrangian formalism and field theory, as well as the integration of ASDYM and twistor theory.

TL;DR: In this article, Liouville et al. constructed an explicit local obstruction to the existence of a Levi-Civita connection within a given projective structure on a surface, which can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of $\Gamma$ or a weighted scalar projective invariant of the projective class.

TL;DR: In this paper, it was shown that finding such metrics reduces to solving a fourth-order integrable partial differance in the signature of a real spinor, and that finding these metrics can be solved by solving a 4-order integral partial.