Integrable perturbations of saddle singularities of rank 0 of integrable Hamiltonian systems

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.
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