Explicit nondegeneracy conditions of KAM tori for the planar N-point vortex systems

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In this paper, we give an explicit nondegeneracy condition for the existence of Kolmogorov-Arnold-Moser (KAM) tori of an N-point vortex system on the plane by using the method of reduction via generalized Jacobi coordinates and matrix theory. Furthermore, by constructing a series of canonical transformations to reduce the degree of freedom of the Hamiltonian, we obtain a new simplified Hamiltonian system. Finally, we give the equivalent relationship between the relative equilibrium point of the original system and the equilibrium point of the new system.