  The main purpose of this paper is to study the finite-dimensional solvable
Lie algebras described in its title, which we call {\em minimal non-${\mathcal
N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent
length $k$, and of nilpotent length $\leq k$, and {\em extreme} Lie algebras,
which have the property that their nilpotent length is equal to the number of
conjugacy classes of maximal subalgebras. We characterise the minimal
non-${\mathcal N}$ Lie algebras in which every nilpotent subalgebra is abelian,
and those of solvability index $\leq 3$.
