Symplectic Classification for Universal Unfoldings of $A_n$ Singularities in Integrable Systems

In the present paper, we obtain real-analytic symplectic normal forms for integrable Hamiltonian systems with $n$ degrees of freedom in small neighborhoods of singular points having the type ``universal unfolding of $A_n$ singularity'', $n\geqslant 1$ (local singularities), as well as in small neighborhoods of compact orbits containing such singular points (semilocal singularities). We also obtain a classification, up to real-analytic symplectic equivalence, of real-analytic Lagrangian foliations in saturated neighborhoods of such singular orbits (semiglobal classification). These corank-one singularities (local, semilocal and semiglobal ones) are structurally stable. It turns out that all integrable systems are symplectically equivalent near their singular points of this type, thus there are no local symplectic invariants. A complete semilocal (respectively, semiglobal) symplectic invariant of the singularity is given by a tuple of $n-1$ (respectively $n-1+\ell$) real-analytic function germs in $n$ variables, where $\ell$ is the number of connected components of the complement of the singular orbit in the fiber. The case $n=1$ corresponds to nondegenerate singularities (of elliptic and hyperbolic types) of one-degree-of-freedom Hamiltonians; their symplectic classifications were known. The case $n=2$ corresponds to parabolic points, parabolic orbits and cuspidal tori.