  The pseudo-potential method is applied to derive diverse propagating electron
hole structures, in a nonthermal or $\kappa$ particle distribution function
background. The associated distribution function Ansatz reproduces the Schamel
distribution of \cite{Schamel2015} in the Maxwellian ($\kappa \rightarrow
\infty$) limit, providing a significant generalization of it for plasmas where
superthermal electrons are ubiquitous, such as space plasmas. The
pseudo-potential and the nonlinear dispersion relation are evaluated. The role
of the spectral index $\kappa$ on the nonlinear dispersion relation is
investigated, in what concerns the wave amplitude for instance. The energy-like
first integral from Poisson's equation is applied to analyze the properties of
diverse classes of solutions: with the absence of trapped electrons, with a
non-analytic distribution of trapped electrons, or with a surplus of trapped
electrons. Special attention is therefore paid to the non-orthodox case where
the electrons distribution function exhibits strong singularities, being
discontinuous or non-analytic.
