  We study a class of partial differential equations (PDEs) in the family of
the so-called Euler-Poincar\'e differential systems, with the aim of developing
a foundation for numerical algorithms of their solutions. This requires
particular attention to the mathematical properties of this system when the
associated class of elliptic operators possesses non-smooth kernels. By casting
the system in its Lagrangian (or characteristics) form, we first formulate a
particles system algorithm in free space with homogeneous Dirichlet boundary
conditions for the evolving fields. We next examine the deformation of the
system when non-homogeneous "constant stream" boundary conditions are assumed.
We show how this simple change at the boundary deeply affects the nature of the
evolution, from hyperbolic-like to dispersive with a non-trivial dispersion
relation, and examine the potentially regularizing properties of singular
kernels offered by this deformation. From the particle algorithm viewpoint,
kernel singularities affect the existence and uniqueness of solutions to the
corresponding ordinary differential equations systems. We illustrate this with
the case when the operator kernel assumes a conical shape over the spatial
variables, and examine in detail two-particle dynamics under the resulting lack
of Lipschitz-continuity. Curiously, we find that for the conically-shaped
kernels the motion of the related two-dimensional waves can become completely
integrable under appropriate initial data. This reduction projects the
two-dimensional system to the one-dimensional completely integrable
Shallow-Water equation [Camassa, R. and Holm, D. D., Phys. Rev. Lett., 71,
1961-1964, 1993], while retaining the full dependence on two spatial dimensions
for the single channel solutions.
