  We study locally compact groups having all dense subgroups (locally) minimal.
We call such groups densely (locally) minimal. In 1972 Prodanov proved that the
infinite compact abelian groups having all subgroups minimal are precisely the
groups $\mathbb Z_p$ of $p$-adic integers. In [31], we extended Prodanov's
theorem to the non-abelian case at several levels. In this paper, we focus on
the densely (locally) minimal abelian groups.
  We prove that in case that a topological abelian group $G$ is either compact
or connected locally compact, then $G$ is densely locally minimal if and only
if $G$ either is a Lie group or has an open subgroup isomorphic to $\mathbb
Z_p$ for some prime $p$. This should be compared with the main result of [9].
Our Theorem C provides another extension of Prodanov's theorem: an infinite
locally compact group is densely minimal if and only if it is isomorphic to
$\mathbb Z_p$. In contrast, we show that there exists a densely minimal,
compact, two-step nilpotent group that neither is a Lie group nor it has an
open subgroup isomorphic to $\mathbb Z_p$.
