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We introduce a new class of connected solvable Lie groups called $H$-group. Namely a $H$-group is a connected solvable Lie group with center $Z$ such that for some $\ell$ in the Lie algebra $h$ of $H$, the symplectic for $B_{\ell }$ on $h/z$ given by $\ell \left(\right[x,y\left]\right)$ is nondegenerate. Moreover, apart form some technical requirements, it will be proved that a connected unimodular Lie group $G$ with center $Z$, such that the center of $G/\mathrm{Rad}G$ is finite, has discrete series if and only if $G$ may be written as $G=H{S}^{\text{'}}$, $H\cap S=Z^0$, where $H$ is a $H$-group with center $Z^0$ and...