  We have developed a new boot-strap method for solving a class of interacting
one-dimensional chiral fermions. The conventional model for interacting
right-moving electrons with spin has an SO(4) symmetry, and can be written as
four interacting Majorana fermions, each with the same velocity. We have found
a method for solving some cases when the velocities of these Majorana fermions
are no longer equal. We demonstrate in some detail the remarkable result that
corrections to the non skeleton self-energy identically vanish for these
models, and this enables us to solve them exactly. For the cases where the
model can be solved by bosonization, our method can be explicitly checked.
However, we are also able to solve some new cases where the excitation spectrum
differs qualitatively from a Luttinger liquid.
  Of particular interest, is the so-called SO(3) model, where a triplet of
Majorana fermions moving at one velocity, interact with a single Majorana
fermion moving at another velocity. We show using our method, that a sharp
bound (or anti-bound) state splits off from the original Luttinger liquid
continuum, cutting off the X-ray singularity to form a broad incoherent
excitation with a lifetime that grows linearly with frequency.
