On an odd dimensional manifold, we define a structure which generalizes several known structures
on almost contact manifolds, namely Sasakian, trans-Sasakian, quasi-Sasakian,
Kenmotsu and cosymplectic structures. This structure, hereinafter called a G.Q.S. manifold, is defined on an
almost contact metric manifold and satisfies an additional condition (1.5). We then consider a codimension-one
distribution on a G.Q.S. manifold. Necessary and
sufficient conditions for the normality of the complemented framed structure on the distribution defined on a G.Q.S
manifold are studied (Th. 3.2). The existence of the foliation on G.Q.S. manifolds and of bundle-like metrics
are also proven. It is shown that under certain circumstances a new foliation arises and its properties are investigated.
Some examples illustrating these results are given in the final part of this paper.