Geodesic
Integrable perturbations of saddle singularities of rank~0 of integrable Hamiltonian systems
Sbornik. Mathematics, Tome 209 (2018) no. 9, pp. 1351-1375

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We study the stability property of singularities of integrable Hamiltonian systems under integrable perturbations. It is known that among singularities of corank $1$, only singularities of complexity $1$ are stable. As it turns out, in the case of two degrees of freedom, there are both stable and unstable singularities of rank $0$ and complexity $2$. A complete list of singularities of saddle-saddle type of complexity $2$ is known and it consists of 39 pairwise non-equivalent singularities. In this paper we prove a criterion for the stability of multi-dimensional saddle singularities of rank $0$ under component-wise perturbations. Using this criterion, in the case of two degrees of freedom, for each of the 39 singularities of complexity $2$ we obtain an answer to the question of whether this singularity is component-wise stable. For a singularity of saddle-saddle type we analyse the connection between the stability property and the characteristics of its loop molecule. Bibliography: 27 titles.
Keywords: integrable Hamiltonian systems, momentum map, nondegenerate singularities, stability. 
@article{SM_2018_209_9_a4,
     author = {A. A. Oshemkov and M. A. Tuzhilin},
     title = {Integrable perturbations of saddle singularities of rank~0 of integrable {Hamiltonian} systems},
     journal = {Sbornik. Mathematics},
     pages = {1351--1375},
     publisher = {mathdoc},
     volume = {209},
     number = {9},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_9_a4/}
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A. A. Oshemkov; M. A. Tuzhilin. Integrable perturbations of saddle singularities of rank~0 of integrable Hamiltonian systems. Sbornik. Mathematics, Tome 209 (2018) no. 9, pp. 1351-1375. http://geodesic.mathdoc.fr/item/SM_2018_209_9_a4/