Abstract. We show that classical Wilczynski–Se-ashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of non-linear ODEs. We explore geometric structures associated with equations that have vanishing generalized Wilczynski invariants and establish relationship of such equations with deformation theory of rational curves on complex algebraic surfaces. 1.

We investigate integrable second order equations of the form F (uxx, uxy, uyy, uxt, uyt, utt) = 0. Familiar examples include the Boyer-Finley equation uxx + uyy = e utt, the potential form of the dispersionless Kadomtsev-Petviashvili (dKP) equation uxt − 1 2 u2 xx = uyy, the dispersionless Hirota equation (α − β)e uxy + (β − γ)e uyt + (γ − α)e utx = 0, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit.

Exterior differential systems for ordinary differential equations

Abstract. We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find all fundamental invariants of a system of third order ODEs and, in particular, determine when a system of third order ODEs is trivializable. As application differential invariants of equations on circles in R n are computed. Key words: geometry of ordinary differential equations; normal Cartan connections 2010 Mathematics Subject Classification: 34A26; 53B15 1

Abstract. We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and co-calibrated homogeneous G2 structure on the seven– dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated G2 structure on SU(2, 1)/U(1). Cuspidal cubics and their higher degree analogues with constant projective curvature are characterised as integral curves of 7th order ODEs. Projective orbits of such curves are shown to be analytic continuations of Aloff–Wallach manifolds, and it is shown that only cubics lift to a complete family of contact rational curves in a projectivised cotangent bundle to a projective plane.

Ordinary differential equations which linearize on differentiation

Paraconformal geometry of nth order ODEs, and exotic holonomy in dimension four

Paraconformal geometry of nth order ODEs, and exotic holonomy in dimension four

Paraconformal geometry of nth order ODEs, and exotic holonomy in dimension four

Paraconformal geometry of nth order ODEs, and exotic holonomy in dimension four