We present a simple procedure to perform the linear stability analysis of a vortex pair equilibrium on a genus zero surface with an arbitrary metric. It consists of transferring the calculations to the round sphere in $\mathbb{R}^3$, with a conformal factor, and exploring the Möbius invariance of the conformal structure, so that the equilibria, seen on the representing sphere, appear in the north/south poles. Three example problems are analyzed: $i)$ For a surface of revolution of genus zero, a vortex pair located at the poles is nonlinearly stable due to integrability. We compute the two frequencies of the linearization. One is for the reduced system, the other is related to the reconstruction. Exceptionally, one of them can vanish. The calculation requires only the local profile at the poles and one piece of global information (given by a quadrature). $ii)$ A vortex pair on a double faced elliptical region, limiting case of the triaxial ellipsoid when the smaller axis goes to zero. We compute the frequencies of the pair placed at the centers of the faces. $iii)$ The stability, to a restricted set of perturbations, of a vortex equilateral triangle located in the equatorial plane of a spheroid, with polar vortices added so that the total vorticity vanishes.

In the class of strictly convex smooth boundaries each of which has no strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the ``normalized'' Mather's $\beta$-function are invariant under $C^\infty$-conjugacies. In contrast, we prove that any two elliptic billiard maps are $C^0$-conjugate near their respective boundaries, and $C^\infty$-conjugate, near the boundary and away from a line passing through the center of the underlying ellipse. We also prove that, if the billiard maps corresponding to two ellipses are topologically conjugate, then the two ellipses are similar.

We study Hamiltonian systems with two degrees of freedom near an equilibrium point, when the linearized system is not semisimple. The invariants of the adjoint linear system determine the normal form of the full Hamiltonian system. For work on stability or bifurcation the problem is typically reduced to a semisimple (diagonalizable) case. Here we study the nilpotent cases directly by looking at the Poisson algebra generated by the polynomials of the linear system and its adjoint.

A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalized Kepler potential. The angular components, on the contrary, are given implicitly by a generally transcendental equation. In the present note, devoted to the previously less studied models with the radial potential of the generalized Kepler type, a new two-parameter family of relevant angular potentials is constructed in terms of elementary functions. For an appropriate choice of parameters, the family reduces to an asymmetric spherical Higgs oscillator.

Quasi-periodic nonconservative perturbations of two-dimensional nonlinear Hamiltonian systems are considered. The definition of a degenerate resonance is introduced and the topology of a degenerate resonance zone is studied. Particular attention is paid to the synchronization process during the passage of an invariant torus through the resonance zone. The existence of so-called synchronization intervals is proved and new phenomena which have to do with synchronization are found. The study is based on the analysis of a pendulum-type averaged system that determines the dynamics near the degenerate resonance phase curve of the unperturbed system.