Classification and new solutions of the Vlasov equation by means of singularity analysis

Singularity analysis, which is an extension of the classical Painlev\'e (P-) test, is applied to classify solutions of the one-dimensional one-component Vlasov-Poisson system and to find its new explicit solutions. As the classical test has been derived for purely differential equations, far-reaching modifications are introduced.
The traditional application of the test is the distinction between the equations, which are integrable and those whose solutions may be chaotic, without solving the equation. The former generically have the P-property, the latter do not, thus the Vlasov equation cannot have the P-property. However the goal of the analysis is classifying and deriving the solutions rather than testing for integrability.
The classification criteria consist of the number and order of the poles in the meromorphic solutions. For the systems, which contain only first-order poles, a method of calculating solutions is given and some new solutions are provided explicitly.
The results may be applied to low-density plasmas as a description of guiding-center dynamics in a very strong magnetic field.