  The spectra of two-dimensional $su(2)$ gauge theories coupled to a single
massless Majorana fermion in integer representations, $J$, are numerically
investigated using the Discrete Light-Cone Hamiltonian. One of our aims is to
explore the possible presence of massless states for $J>2$ in spite of the
absence of a continuous symmetry. After comparing to existing results for $J=1$
(adjoint fermions), we present results for $J=2,3,4$. As expected, for $J=2$
there are no massless states but in contrast to the $J=1$ theory, the lightest
state is a boson. We find exact massless modes in the bosonic and fermionic
sector for all values of total momentum for $J=3$ and $J=4$ and, in each
sector, the number of massless modes grows with the value of the total
momentum. In addition to the spectrum, we present results on the particle
number and momentum fraction distributions and argue for a separation of bulk
states from edge states.
