Complete integrability versus symmetry

Contact complete integrability

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and coLegendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the exis...

Symmetry, singularities and integrability in complex dynamics V: Complete symmetry groups of certain relativistic spherically symmetric systems

Abstract: We show that the concept of complete symmetry group introduced by Krause (J Math Phys 35 (1994) 5734-5748) in the context of the Newtonian Kepler problem has wider applicability, extending to the relativistic context of the Einstein equations describing spherically symmetric bodies with certain conformal Killing symmetries. We also provide a simple demonstration of the nonuniqueness o...

One Symmetry Implies Symmetry-Integrability for Scalar Evolution Equations

We prove two general results on generalized symmetries for equations of the form ut = um + f(u, u1, . . . , um−1), where f is a formal (differential) power series starting with terms that are at least quadratic. The first result states that any higher order symmetry must be also a differential polynomial if f is a differential polynomial of order less than m − 1. The method is to estimate the o...

Symmetry and integrability in Hamiltonian normal forms

One Symmetry Does Not Imply Integrability

We show that Bakirov’s counterexample (which had been checked by computeralgebra methods up till order 53) to the conjecture that one nontrivial symmetry of an evolution equation implies infinitely many is indeed a counterexample. To prove this we use the symbolic method of Gel’fand-Dikii and p-adic analysis. We also formulate a conjecture to the effect that almost all equations in the family c...