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A singular Poincaré lemma. (English) Zbl 1078.58007
The authors prove the following result: let $$g_{1},\dots , g_{r}$$ be a set of germs of smooth functions on $$(\mathbb{R}^{2n}, 0), r\leq n$$ fulfilling the commutation relations: $$X_{i}(g_{j})=X_{j}(g_{i})$$ for any $$i,j \in \{ 1,2,\dots ,r \}$$, where $$\{ X_{1},\dots , X_{r} \}$$ is a basis of a Cartan subalgebra of $$\text{sp}(2n, \mathbb{R})$$. Then there exists a germ of a smooth function $$G$$ and $$r$$ germs of smooth functions $$f_{i}$$ such that: (i) $$X_{j}(f_{i})=0$$ for any $$i,j \in \{ 1,2,\dots ,r \}$$; (ii) $$g_{i}=f_{i} + X_{i}(G)$$ for any $$i \in \{ 1,2,\dots ,r \}$$.
Some applications are also nicely pointed out.

MSC:
 58C25 Differentiable maps on manifolds 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 53D05 Symplectic manifolds, general 58A10 Differential forms in global analysis
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