  We introduce an extension of the \ell-reduced KP hierarchy, which we call the
\ell-Bogoyavlensky hierarchy. Bogoyavlensky's 2+1-dimensional extension of the
KdV equation is the lowest equation of the hierarchy in case of \ell=2. We
present a group-theoretic characterization of this hierarchy on the basis of
the 2-toroidal Lie algebra sl_\ell^{tor}. This reproduces essentially the same
Hirota bilinear equations as those recently introduced by Billig and Iohara et
al. We can further derive these Hirota bilinear equation from a Lax formalism
of the hierarchy.This Lax formalism also enables us to construct a family of
special solutions that generalize the breaking soliton solutions of
Bogoyavlensky. These solutions contain the N-soliton solutions, which are
usually constructed by use of vertex operators.
