  We extend the notion of essential deformations from the case of the Iwasawa
manifold, for which they were introduced recently by the first-named author, to
the general case of page-$1$-$\partial\bar\partial$-manifolds that were jointly
introduced very recently by all three authors. We go on to obtain an analogue
of the unobstructedness theorem of Bogomolov, Tian and Todorov for
page-$1$-$\partial\bar\partial$-manifolds. As applications of this discussion,
we study the small deformations of certain Nakamura solvmanifolds and
reinterpret the cases of the Iwasawa manifold and its $5$-dimensional analogue
from this standpoint.
