On pro-solvable Lie algebras with maximal pro-nilpotent ideals ${\mathfrak{m}}_0$ and ${\mathfrak{m}}_2$

The paper is devoted to the study of pro-solvable Lie algebras whose maximal pro-nilpotent ideal is either ${\mathfrak{m}}_0$ or ${\mathfrak{m}}_2$. Namely, we describe such Lie algebras and establish their completeness. Triviality of the second cohomology group for one of the obtained algebra is established.