The author classifies left invariant Einstein metrics on some class of 7-dimensional solvable Lie groups using the variational approach, proposed by J. Lauret. The problem reduces to the classification of some solvable 7-dimensional Einstein metric Lie algebras $(\frak g, g)$, that is, a Lie algebra $\frak g$ together with an Euclidean metric $g$ which satisfy the Einstein equation.
More precisely, let $(\frak n ,g)$ be a metric nilpotent Lie algebra and $d$ a $g$-symmetric derivation of $\frak n$. Then the solvable metric Lie algebra $(\frak g , \widetilde g )$, where $\frak g = \Bbb R d + \frak n$ is a semidirect sum and $\widetilde g$ is a natural extension of the metric $g$ such that $\widetilde g(d,d) =1$, $\widetilde g(d,\frak n) =0$ is called a metric rank one solvable extension of $(\frak n, g)$. It is known that if $(\frak g , \tilde g)$ is an Einstein metric Lie algebra which contains $(\frak n, g)$ as a codimension one (metric) nilpotent ideal, then $(\frak g, \widetilde g)$ is a uniquely defined metric rank one solvable extension of $(\frak g, g)$.
The author classifies all metric rank one extensions of 6-dimensional metric nilpotent Lie algebras $(\frak n, g)$ which are Einstein metric Lie algebras. She finds 34 metric nilpotent 6-dimensional Lie algebras, which have such extension.