  The average energy of the Ising spin glass is known to have no singularity
along a special line in the phase diagram although there exists a critical
point on the line. This result on the model with uncorrelated disorder is
generalized to the case with correlated disorder. For a class of correlations
in disorder that suppress frustration, we show that the average energy in a
subspace of the phase diagram is expressed as the expectation value of a local
gauge variable of the $Z_2$ gauge Higgs model, from which we prove that the
average energy has no singularity although the subspace is likely to have a
phase transition on it. This is a generalization of the result for uncorrelated
disorder. Though it is difficult to obtain an explicit expression of the energy
in contrast to the case of uncorrelated disorder, an exact closed-form
expression of a physical quantity related to the energy is derived in three
dimensions using a duality relation. Identities and inequalities are proved for
the specific heat and correlation functions.
