  In this work we consider periodic spherically symmetric metrics of constant
positive scalar curvature on the n-dimensional cylinder called pseudo-cylindric
metrics. These metrics belong to the conformal class $[g_0]$ of the Riemannian
product $S^1\times S^{n-1}$ : a circle of length $T$ crossed with the
(n-1)-dimensional standard sphere. Such metrics have a harmonic Riemannian
curvature and a non parallel Ricci tensor, except for the cylindric one. Thus,
it appears a natural link between them and the Derdzinski metrics which are
warped product and classify a family of Riemannian manifolds. These two
families actually differ by conformal transformations. Moreover, we are
interested in the multiplicity problem of the pseudo-cylindric metrics in
$[g_0]$. We also study the existence problem and the number of the Derdzinski
metrics. Furthermore, we prove that the pseudo-cylindric metrics may be
expressed in terms of elliptic functions for the dimension $n = 3,4$ and 6
only, and in terms of automorphic functions for any other dimension. This fact
allows us to give new bounds for global estimates. Finally, we examine the
curvature of the asymptotic pseudo-cylindric metrics which are complete
singular Yamabe metrics on the standard sphere punctured of $k$ points. We show
that any of such (non trivial) metric has a non parallel Ricci tensor.
