  This paper concerns with some of the results related to the singular
solutions of certain types of non-linear integrable differential equations
(NIDE) and behavior of the singularities of those equations. The approach
heavily relies on the Method of Operator Identities which proved to be a
powerful tool in different areas such as interpolation problems, spectral
analysis, inverse spectral problems, dynamic systems, non-linear equations. We
formulate and solve a number of problems (direct and inverse) related to the
singular solutions of sinh-Gordon, non-linear Schr\"{o}dinger and modified
Korteweg - de Vries equations. Dynamics of the singularities of these solutions
suggests that they can be interpreted in terms of particles interacting through
the fields surrounding them. We derive differential equations describing the
dynamics of the singularities and solve some of the related problems. The
developed methodologies are illustrated by numerous examples.
