  In this paper, we explore the structure of the normal Sally modules of rank
one with respect to an $m$-primary ideal in a Nagata reduced local ring which
is not necessary Cohen-Macaulay. As an application of this result, when the
base ring is Cohen-Macaulay analytically unramified, the extremal bound on the
first normal Hilbert coefficient leads to the Cohen-Macaulayness of the
associated graded rings with respect to a normal filtration.
