  The extended flow equations of a new $Z_N$-Toda hierarchy which takes values
in a commutative subalgebra $Z_N$ of $gl(N,\mathbb C)$ is constructed.
Meanwhile we give the Hirota bilinear equations and tau function of this new
extended $Z_N$-Toda hierarchy(EZTH). Because of logarithm terms, some extended
Vertex operators are constructed in generalized Hirota bilinear equations which
might be useful in topological field theory and Gromov-Witten theory. Meanwhile
the Darboux transformation and bi-hamiltonian structure of this hierarchy are
given. From hamiltonian tau symmetry, we give another different tau function of
this hierarchy with some unknown mysterious connections with the one defined
from the point of Sato theory.
