  Vlasov-Maxwell equilibria are described by the self-consistent solutions of
the time-independent Maxwell equations for the real-space dynamics of
electromagnetic fields, and the Vlasov equation for the phase-space dynamics of
particle distributions in a collisionless plasma. These two systems
(macroscopic and microscopic) are coupled via the source terms in Maxwell's
equations, which are sums of velocity-space 'moment' integrals of the particle
distribution function. This paper considers 'the inverse problem for
collisionless equilibria' (IPCE), viz. "given information regarding the
real-space/macroscopic configuration of a specific collisionless plasma
equilibrium, what self-consistent equilibrium distributions exist?" We develop
the constants of motion approach to IPCE using the assumptions of a 'modified
Maxwellian' distribution function, and a strictly neutral and spatially
one-dimensional plasma. In such circumstances, IPCE formally reduces to the
inversion of Weierstrass transformations (Bilodeau, 1962), such as those
transformations that feature in the initial value problem for the
heat/diffusion equation. We discuss the various mathematical conditions that a
candidate solution of IPCE must satisfy. One method that can be used to invert
the Weierstrass transform is expansions in Hermite polynomials. Building on the
results of Allanson et al. (2016), we establish under what circumstances a
solution obtained by these means converges, and allows velocity moments of all
orders. Ever since the seminal work by Bernstein et al. (1957), on 'stationary'
electrostatic plasma waves, the necessary quality of non-negativity has been
noted as a feature that any candidate solution of IPCE will not $\textit{a
priori}$ satisfy. We also discuss this problem in the context of for our
formalism, for magnetised plasmas.
