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For a sequence of the naturally graded quasi-filiform Leibniz algebra ofsecond type $\mathcal{L}^2$ introduced by Camacho, G\'{o}mez, Gonz\'{a}lez andOmirov, all possible right and left solvable indecomposable extensions over thefield $\Bbb R$ are constructed so that the algebra serves as the nilradical ofthe corresponding solvable Leibniz algebras we find in the paper.

Solvable extensions of the naturally graded quasi-filiform Leibniz algebra of second type $\mathcal{L}^2$