  It is shown that variants of the HI methods could yield objects closely
connected to the classical Banach spaces. Thus we present a new $c_0$ saturated
space, denoted as $\mathfrak{X}_0$, with rather tight structure. The space
$\mathfrak{X}_0$ is not embedded into a space with an unconditional basis and
its complemented subspaces have the following structure. Everyone is either of
type I, namely, contains an isomorph of $\mathfrak{X}_0$ itself or else is
isomorphic to a subspace of $c_0$ (type II). Furthermore for any analytic
decomposition of $\mathfrak{X}_0$ into two subspaces one is of type I and the
other is of type II. The operators of $\mathfrak{X}_0$ share common features
with those of HI spaces.
