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Let $\mathrm{Lie} \left(n, k\right)$ denote the class of all $n$-dimensionalreal solvable Lie algebras having $k$-dimensional derived ideal ($1 \leqslant k\leqslant n-1$). In 1993, the class $\mathrm{Lie} \left(n, 1\right)$ wascompletely classified by Sch\"obel \cite{Sch93}. In 2016, Vu A. Le et al.\cite{VHTHT16} considered the class $\mathrm{Lie} \left(n, n-1\right)$ andclassified its subclass containing all the algebras having 1-codimensionalcommutative derived ideal. One subclass in {\Li} was firstly considered andincompletely classified by Sch\"obel \cite{Sch93} in 1993. Later, Janisse alsogave an incomplete classification of {\Li} and published as a scientific report\cite{Jan10} in 2010. In this paper, we set up a new approach to study theclassifying problem of classes {\Li} as well as {\li} and present the newcomplete classification of {\Li} in the combination with the well-knownEberlein's result of 2-step nilpotent Lie algebras from\cite[p.\,37--72]{Ebe03}. The paper will also classify a subclass of {\li} andwill point out missings in Sch\"obel \cite{Sch93}, Janisse \cite{Jan10},Mubarakzyanov \cite{Mub63a} as well as revise an error of Morozov \cite{Mor58}.

On the classifying problem for the class of real solvable Lie algebras having 2-dimensional or 2-codimensional derived ideal