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In this paper solvable Leibniz algebras with naturally graded non-Lie$p$-filiform $(n-p\geq4)$ nilradical and with one-dimensional complementedspace of nilradical are described. Moreover, solvable Leibniz algebras withabelian nilradical and extremal (minimal, maximal) dimensions of complementedspace nilradical are studied. The rigidity of solvable Leibniz algebras withabelian nilradical and maximal dimension of its complemented space is proved.

Solvable Leibniz algebras with naturally graded non-Lie $p$-filiform nilradicals