Normal forms for superintegrable systems at singular points and their perturbation problems

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The purpose of this research is to understand the global structure of solutions for the so-called superintegrable systems admitting integrals the number of which is greater than the degrees of freedom. We generalized Liouville-Arnold theorem which is the fundamental theorem for integrable systems. Namely, under some additional conditions on resonances, we showed the existence of special coordinates in a neighbourhood of singularities of the map defined by those integrals so that the system can be solved explicitly in those coordinates.Moreover, we defined superintegrability for general vector fields and showed that such a superintegrable vector field can be solved explicitly in a neighbourhood of some type of equilibrium point by obtaining its convergent normal form.