  In this paper, we firstly construct a weakly coupled Toda lattices with
indefinite metrics which consist of $2N$ different coupled Hamiltonian systems.
Afterwards, we consider the iso-spectral manifolds of extended tridiagonal
Hessenberg matrix with indefinite metrics what is an extension of a strict
tridiagonal matrix with indefinite metrics. For the initial value problem of
the extended symmetric Toda hierarchy with indefinite metrics, we introduce the
inverse scattering procedure in terms of eigenvalues by using the Kodama's
method. In this article, according to the orthogonalization procedure of
Szeg\"{o}, the relationship between the $\tau$-function and the given Lax
matrix is also discussed. We can verify the results derived from the
orthogonalization procedure with a simple example. After that, we construct a
strongly coupled Toda lattices with indefinite metrics and derive its tau
structures. At last, we generalize the weakly coupled Toda lattices with
indefinite metrics to the $Z_{n}$-Toda lattices with indefinite metrics.
