We study certain classes of integrable nonlinear differential equations related to the type symmetric spaces. Our main examples concern equations related to A.III-type symmetric spaces. We use the Cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic Lax operators to this symmetric space. Next we outline the spectral theory of the reduced Lax operator $L$ and construct its fundamental analytic solutions. Analyzing the Wronskian relations we introduce the `squared solutions' of $L$ and derive the recursion operators by three different methods.

In this paper, we calculate the frequency of small oscillations of a ring confined to a plane that, along with a fixed ring of the same size, supports a soap film catenoid. The restoring force is provided by the film's surface tension. We assume that the soap film is massless and continuously assumes the shape of minimal area. Mathematically, the problem is equivalent to calculating the second derivative of the total area with respect to the displacement of the ring. The calculus of moving surfaces is extensively used in the presented calculation.

Star products parametrized by complex matrices are defined. Especially commutative associative star products are treated, and star exponentials with respect to these star products are considered. Jacobi's theta functions are given as infinite sums of star exponentials. As application, several concrete identities are obtained by properties of the star exponentials.

We formulate the equation of motion of a charged particle in a Riemannian manifold with a closed two form. Since a two-step nilpotent Lie group has natural left-invariant closed two forms, it is natural to consider the motion of a charged particle in a simply connected two-step nilpotent Lie groups with a left invariant metric. We study the behavior of the motion of a charged particle in the above spaces.

The Baily-Borel compactification $\widehat{{\mathbb B} / \Gamma}$ of an arithmetic ball quotient admits projective embeddings by $\Gamma$-modular forms of sufficiently large weight. We are interested in the target and the rank of the projective map $\Phi$, determined by $\Gamma$-modular forms of weight one. This paper concentrates on the finite $H$-Galois quotients ${\mathbb B} / \Gamma _H$ of a specific ${\mathbb B} / \Gamma _{-1}^{(6,8)}$, birational to an abelian surface $A_{-1}$. Any compactification of ${\mathbb B} / \Gamma _H$ has non-positive Kodaira dimension. The rational maps $\Phi ^H$ of $\widehat{{\mathbb B} / \Gamma _H}$ are studied by means of the $H$-invariant abelian functions on $A_{-1}$.

We study the linear part of the monodromy of completely integrable Hamiltonian systems via Bohr--Sommerfeld Geometric Quantization. We relate monodromy to the ambiguity in the choice of the pre--quantum connection and to the action of the (connected component of the) gauge group.