In this paper we study some geometric properties of surfaces in the Heisenberg group, $\mathcal{H}_{3}\!.$ We obtain, using the Gauss map for Lie groups, a partial classification of minimal graphs in $\mathcal{H}_{3}\!.$

We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator $L$. The Lax representation is $\mathbb{Z}_2\times\mathbb{Z}_2$ reduced and can be naturally associated with the symmetric space ${\rm SU(3)/S(U(1)\times U(2))}$. The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the $N$-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one-soliton solutions of NLEEs with even and odd dispersion laws have different properties. In particular, the one-soliton solutions for NLEEs with even dispersion laws are not traveling waves while their velocities and amplitudes are time dependent. Calculating the asymptotics of the $N$-soliton solutions for $t\to\pm\infty$ we analyze the interactions of quadruplet solitons.

We show that the MIC-Kepler probrem is simply solved via the phase-space formulation of non-relativistic quantum mechanics. The MIC-Kepler problem is the Hamiltonian system behind the hydrogen atom subjected to the influence of the Dirac’s magnetic monopole field and the square inverse centrifugal potential force besides the Coulomb’s potential force. We get the energy spectrum of the bound states explicitly and construct the Green’s functions for $E ˂ 0$ by means of the Moyal product, which is one of the ‘star’ products denoted by ‘$\ast$’.

We give geometric picture for the Recursion Operators related to the Caudrey-Beals-Coifman linear problem and the Nonlinear Evolution Equations associated to it in the presence of $\mathbb{Z}_p$ reductions of Mikhailov type.