The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows:

When the curvature of the complex space form is nonzero, there is a 1-parameter family of such hypersurfaces. Specifically, for each one-parameter subgroup of the isometry group of the complex space form, there is an essentially unique example that is invariant under this one-parameter subgroup.

On the other hand, when the curvature of the space form is zero, i.e., when the space form is $\mathbb{C}^2$ with its standard metric, there is an additional 'exceptional' example that has no continuous symmetries but is invariant under a lattice of translations. Up to isometry and homothety, this is the unique example with no continuous symmetries.

We extend the results of [11] on embedded CR manifolds of CR dimension and codimension two to abstract partially integrable almost CR manifolds. We prove that points on such manifolds fall into three different classes, two of which (the hyperbolic and the elliptic points) always make up open sets. We prove that manifolds consisting entirely of hyperbolic (respectively elliptic) points admit canonical Cartan connections. More precisely, these structures are shown to be exactly the normal parabolic geometries of types $(PSU(2, 1) \times PSU(2, 1), B \times B)$, respectively $(PSL(3, \mathbb{C}), B)$, where $B$ indicates a Borel subgroup. We then show how general tools for parabolic geometries can be used to obtain geometric interpretations of the torsion part of the harmonic components of the curvature of the Cartan connection in the elliptic case.

In the context of the cohomological deformation theory, infinitesimal description of one-parametric families of Bäcklund transformations of special type including classical examples is given. It is shown that any family of such a kind evolves in the direction of a nonlocal symmetry shadow in the sense of [9].

The coupled KdV-mKdV system arises as the classical part of one of superextensions of the KdV equation. For this system, we prove its complete integrability, i.e., existence of a recursion operator and of infinite series of symmetries.

In this article we first present a family of Cayley transforms of a homogeneous Siegel domain. We then give characterizations of symmetric Siegel domains among homogeneous Siegel domains in terms of norm equalities involving the Cayley transforms. Applications of these characterizations to analysis on Siegel domains (the Berezin transforms and the Poisson kernel) are also exhibited.

The structure group and the involutive differential system that characterize the pseudo-group of contact transformations on a jet space are determined.

We study the problem of lifting analytic actions of a Lie group $G$ to a non-split complex analytic supermanifold $(M, \mathcal{O})$ from its retract $(M, \mathcal{O}_{\mathrm{gr}})$. In the case when $G$ is compact (or complex reductive), two criteria for lifting a Lie group action are found. The first one is invariance of the Čech 1-cocycle with values in a special automorphism sheaf of $(M, \mathcal{O}_{\mathrm{gr}})$ determining the non-split supermanifold $(M, \mathcal{O})$, while the second one is invariance of a certain differential form of a special kind which can also be viewed as a global derivation of a sheaf of differential forms on $M$. If the action is transitive on $M$, then the second criterion allows to reduce the lifting problem to the study of invariants of a finite dimensional linear representation.

The second author introduced Schwarzian derivatives for 3-dimensional contact transformations in [Sat]. Our purposes of this paper are firstly to investigate the fundamental properties of the contact Schwarzian derivatives of 3-dimensional contact transformations that are satisfied by the usual Schwarzian derivatives, secondly to consider systems of linear PDE's with contact Schwarzian derivatives as coefficients and their integrability conditions, and finally to reconstruct the contact transformation from the solutions of the systems of linear PDE's. We obtain the necessary and sufficient condition for functions to be contact Schwarzian derivatives of a 3-dimensional contact transformation.

We give a brief review on Weyl manifolds and thier Poincaré-Cartan classes. A Weyl manifold is a Weyl algebra bundle over a symplectic manifold which is a geometrization of deformation quantization and the Poincaré-Cartan class is a complete invariant of Weyl manifolds.

We introduce a concept of a contact Weyl manifold, which is a contact algebra bundle over a symplectic manifold containing a Weyl manifold as a subbundle. We show the existence of contact Weyl manifolds for a symplectic manifold.

We construct a connection on a contact Weyl manifold which gives a Fedosov connection when it is restricted to a Weyl manifold. With the help of the connection, we show that the cohomology class given by the curvature of Fedosov connection coincides with the Poincaré-Cartan class.