Twistor geometry of a pair of second order ODEs

We discuss the twistor correspondence between path geometries in three dimensions with vanishing Wilczynski invariants and anti-self-dual conformal structures of signature $(2, 2)$. We show how to reconstruct a system of ODEs with vanishing invariants for a given conformal structure, highlighting the Ricci-flat case in particular. Using this framework, we give a new derivation of the Wilczynski invariants for a system of ODEs whose solution space is endowed with a conformal structure. We explain how to reconstruct the conformal structure directly from the integral curves, and present new examples of systems of ODEs with point symmetry algebra of dimension four and greater which give rise to anti--self--dual structures with conformal symmetry algebra of the same dimension. Some of these examples are $(2, 2)$ analogues of plane wave space--times in General Relativity. Finally we discuss a variational principle for twistor curves arising from the Finsler structures with scalar flag curvature.Comment: Final version to appear in the Communications in Mathematical Physics. The procedure of recovering a system of torsion-fee ODEs from the heavenly equation has been clarified. The proof of Prop 7.1 has been expanded. Dedicated to Mike Eastwood on the occasion of his 60th birthda

An algebraic method of classification of S-integrable discrete models

A method of classification of integrable equations on quad-graphs is discussed based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators of the ring generators. For the generic case this function grows exponentially. Examples show that for integrable equations it grows slower. We propose a classification scheme based on this observation.Comment: 11 pages, workshop "Nonlinear Physics. Theory and Experiment VI", submitted to TM

Explicit differential characterization of the Newtonian free particle system in m > 1 dependent variables

In 1883, as an early result, Sophus Lie established an explicit necessary and sufficient condition for an analytic second order ordinary differential equation y_xx = F(x,y,y_x) to be equivalent, through a point transformation (x,y) --> (X(x,y), Y(x,y)), to the Newtonian free particle equation Y_XX = 0. This result, preliminary to the deep group-theoretic classification of second order analytic ordinary differential equations, was parachieved later in 1896 by Arthur Tresse, a French student of S. Lie. In the present paper, following closely the original strategy of proof of S. Lie, which we firstly expose and restitute in length, we generalize this explicit characterization to the case of several second order ordinary differential equations. Let K=R or C, or more generally any field of characteristic zero equipped with a valuation, so that K-analytic functions make sense. Let x in K, let m > 1, let y := (y^1, ..., y^m) in K^m and let y_xx^j = F^j(x,y,y_x^l), j = 1,...,m be a collection of m analytic second order ordinary differential equations, in general nonlinear. We provide an explicit necessary and sufficient condition in order that this system is equivalent, under a point transformation (x, y^1, ..., y^m) --> (X(x,y), Y^1(x,y),..., Y^m(x, y)), to the Newtonian free particle system Y_XX^1 = ... = Y_XX^m = 0. Strikingly, the (complicated) differential system that we obtain is of first order in the case m > 1, whereas it is of second order in S. Lie's original case m = 1.Comment: 76 pages, no figur