  We study the six-dimensional solvmanifolds that admit complex structures of
splitting type classifying the underlying solvable Lie algebras. In particular,
many complex structures of this type exist on the Nakamura manifold $X$, and
they allow us to construct a countable family of compact complex
non-$\partial\overline\partial$ manifolds $X_k$, $k\in\mathbb{Z}$, that admit a
small holomorphic deformation $\{(X_{k})_{t}\}_{t\in\Delta_k}$ satisfying the
$\partial\overline\partial$-Lemma for any $t\in\Delta_k$ except for the central
fibre. Moreover, a study of the existence of special Hermitian metrics is also
carried out on six-dimensional solvmanifolds with splitting-type complex
structures.
