The main purpose of this paper is to investigate commuting flows and integrable systems on the configuration spaces of planar linkages. Our study leads to the definition of a natural volume form on each configuration space of planar linkages, the not
ion of cross products of integrable systems, and also the notion of multi-Nambu integrable systems. The first integrals of our systems are functions of Bott-Morse type, which may be used to study the topology of configuration spaces. Dedicated to Anatoly T. Fomenko on the occasion of his 75th birthday
In this paper we prove the existence of a simultaneous local normalization for couples $(X,mathcal{G})$, where $X$ is a vector field which vanishes at a point and $mathcal{G}$ is a singular underlying geometric structure which is invariant with respe
ct to $X$, in many different cases: singular volume forms, singular symplectic and Poisson structures, and singular contact structures. Similarly to Birkhoff normalization for Hamiltonian vector fields, our normalization is also only formal, in general. However, when $mathcal{G}$ and $X$ are (real or complex) analytic and $X$ is analytically integrable or Darboux-integrable then our simultaneous normalization is also analytic. Our proofs are based on the toric approach to normalization of dynamical systems, the toric conservation law, and the equivariant path method. We also consider the case when $mathcal{G}$ is singular but $X$ does not vanish at the origin.
In this communication, we show that both infinite-dimension
