  This paper is devoted to the complete algebraic and geometric classification
of complex $4$-dimensional nilpotent weakly associative, complex
$4$-dimensional symmetric Leibniz algebras, and complex $5$-dimensional
nilpotent symmetric Leibniz algebras. In particular, we proved that the variety
of complex $4$-dimensional symmetric Leibniz algebras has no
Vergne--Grunewald--O'Halloran Property (there is an irreducible component
formed by only nilpotent algebras), but on the other hand, it has Vergne
Property (there are no rigid nilpotent algebras).
