We show the change of coordinates that maps the maximally symmetric $(2,3,5)$-distribution given by solutions to the $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy equation to the flat Cartan distribution. This establishes the local equivalence between the maximally symmetric $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy distribution and the flat Cartan or Hilbert-Cartan distribution. We give the set of vector fields parametrised by solutions to the $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy equation and the corresponding Ricci-flat conformal scale that bracket-generate...

A differential 1-form on a $(2k+1)$-dimensional manifolds $M$ defines a singular contact structure if the set $S$ of points where the contact condition is not satisfied, $S=\{p\in M:(\omega \wedge {\left(d\omega \right)}^k\left(p\right)=0\}$, is nowhere dense in $M$. Then $S$ is a hypersurface with singularities and the restriction of $\omega$ to $S$ can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation $\left(\omega \right)$ generated by $\omega$ is determined, up to a diffeomorphism, by its restriction to $S$, if we eliminate certain degenerated singularities...

In this work we consider a class of germs of singularities of integrable 1-forms in $R^n$ which are structurally stable in class $C^r$ ($r\ge 2$ if $n=3$, $r\ge 4$ if $n\ge 4$), whose 1-jet is zero at the singularity. In this class the stability depends essentially on the fact that the perturbations allowed are integrable.

We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of a 𝕂-analytic curve is a finite-dimensional vector space. We also show that the action of local diffeomorphisms preserving the quasi-homogeneous curve on this vector space is determined by the infinitesimal action of...

We introduce the concept of conserved current variationally associated with locally variational invariant field equations. The invariance of the variation of the corresponding local presentation is a sufficient condition for the current beeing variationally equivalent to a global one. The case of a Chern-Simons theory is worked out and a global current is variationally associated with a Chern-Simons local Lagrangian.

Let $G:{ℂ}^n\times {ℂ}^r\to ℂ$ be a holomorphic family of functions. If $\Lambda \subset {ℂ}^n\times {ℂ}^r$, ${\pi }_r:{ℂ}^n\times {ℂ}^r\to {ℂ}^r$ is an analytic variety then   $Q_{\Lambda }\left(G\right)=(x,u)\in {ℂ}^n\times {ℂ}^r:G(·,u)hasacriticalpointin\Lambda \cap {\pi }_r^{-1}\left(u\right)$is a natural generalization of the bifurcation variety of G. We investigate the local structure of $Q_{\Lambda }\left(G\right)$ for locally trivial deformations of $\Lambda ₀={\pi }_r^{-1}\left(0\right)$. In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.