  We study the quantum Hamilton-Jacobi (QHJ) equation of the recently obtained
exactly solvable models, related to the newly discovered exceptional
polynomials and show that the QHJ formalism reproduces the exact eigenvalues
and the eigenfunctions. The fact that the eigenfunctions have zeros and poles
in complex locations leads to an unconventional singularity structure of the
quantum momentum function $p(x)$, the logarithmic derivative of the wave
function, which forms the crux of the QHJ approach to quantization. A
comparison of the singularity structure for these systems with the known
exactly solvable and quasi-exactly solvable models reveals interesting
differences. We find that the singularities of the momentum function for these
new potentials lie between the above two distinct models, sharing similarities
with both of them. This prompted us to examine the exactness of the
supersymmetric WKB (SWKB) quantization condition. The interesting singularity
structure of $p(x)$ and of the superpotential for these models has important
consequences for the SWKB rule and in our proof of its exactness for these
quantal systems.
