On several varieties of Leibniz algebras

The paper is devoted to two new results concerning varieties of Leibniz algebras. In case of prime characteristic $p$ we construct an example of a non-nilpotent variety of Leibniz algebras with Engel condition. In case of field of characteristic zero we obtain a new result concerning the space of multilinear components of the variety of left-nilpotent Leibniz algebra of class three.