Since the last two decades, several differential operators appeared in connection with the so-called oscillatory geometry. These operators act on sections of infinite rank vector bundles. Definitions of the oscillatory representation, metaplectic structure, oscillatory Dirac operator, as well as some necessary fundamental results in the analysis in $C^*$-Hilbert bundles are recalled here. These results are used for a description of the kernel of a certain second order differential operator arising from oscillatory geometry and the cohomology groups of the de Rham complex of exterior forms with values in the oscillatory representation.

We derive parametrizations of the Delaunay constant mean curvature surfaces of revolution that follow directly from parametrizations of the conics that generate these surfaces via the corresponding roulette. This uniform treatment exploits the natural geometry of the conic (parabolic, elliptic or hyperbolic) and leads to simple expressions for the mean and Gaussian curvatures of the surfaces as well as the construction of new surfaces.

In this article we suggest a new method, partially based on earlier works of Wohlhart [15], Mladenova and Mladenov [11], Brezov et al [3], that resolves the generalized Euler decomposition problem (about arbitrary axes) using a system of quadratic equations. The main contribution made here is that we manage to decouple this system and express the solutions independently in a compact covariant form. We apply the same technique to the Lorentz group in $2+1$ dimensions and discuss certain complications related to the presence of isotropic directions in $\mathbb{R}^{2,1}$.

Lie symmetries of $K(m,n)$ equations with time-dependent coefficients are classified. Group classification is presented up to widest possible equivalence groups, the usual equivalence group of the whole class for the general case and the conditional equivalence groups for special values of the exponents $m$ and $n$. Examples on reduction of $K(m,n)$ equations (with initial and boundary conditions) to nonlinear ordinary differential equations (with initial conditions) are presented.

We consider the hierarchies of Nonlinear Evolution Equations related to auxiliary problem of Caudrey-Beals-Coifman type. We give a proof that the conservation laws for these equations have local densities based on the theory of the generating operators related to the Caudrey-Beals-Coifman linear problem.