This article is a brief description, without proofs, of recent results by its author, both alone and with co-workers [R. Brouzet, P. Molino and the author, Indag. Math., New Ser. 4, No. 3, 269-296 (1993; Zbl 0793.58015), the author, Manuscr. Math. 82, No. 3-4, 349-362 (1994; Zbl 0807.53028) and C. R. Acad. Sci., Paris, Sér. I 319, No. 5, 471-474 (1994; Zbl 0817.53006)]. The context of this work is an even-dimensional manifold \(M\) with a pair of compatible symplectic forms, \(\omega\), \(\omega'\). In this case, with some regularity assumptions there is a natural class of Hamiltonian vector fields, the typical orbits of which are constrained to lie on tori of dimension half that of \(M\). The results described in the article concern standard forms (in local coordinates) for \(\omega\) and \(\omega'\), as well as applications to the setting where there is a fibration of \(M\) that is Lagrangian with respect to both of the forms.
For the entire collection see [Zbl 0826.00023].