  Methods for constructing synthetic multidimensional electron hole equilibria
without using particle simulation are investigated. Previous approaches have
various limitations and approximations that make them unsuitable within the
context of expected velocity diffusion near the trapped-passing boundary. An
adjustable model of the distribution function is introduced that avoids
unphysical singularities there, and yet is sufficiently tractable analytically
to enable prescription of the potential spatial profiles. It is shown why
simple models of the charge density as being a function only of potential
cannot give solitary multidimensional electron holes, in contradiction of prior
suppositions. Fully self-consistent axisymmetric electron holes in the
drift-kinetic limit of electron motion (negligible gyro-radius) are constructed
and their properties relevant to observational interpretation and
finite-gyro-radius theory are discussed.
