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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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