  The paper aims to investigate the classification problem of low dimensional
complex none Lie filiform Leibniz algebras. There are two sources to get
classification of filiform Leibniz algebras. The first of them is the naturally
graded none Lie filiform Leibniz algebras and the another one is the naturally
graded filiform Lie algebras. Here we do consider Leibniz algebras appearing
from the naturally graded none Lie filiform Leibniz algebras. This class can be
splited into two subclasses. However, isomorphisms within each class there were
not investigated. Before U.D.Bekbaev and I.S.Rakhimov suggested an approach to
the isomorphism problem in terms of invariants. This paper presents an
implementation of their result in low dimensional cases. Here we give the
complete classification of complex none Lie filiform Leibniz algebras in
dimensions at most 8 from the first class of the above mentioned result and
give a hypothetic formula for the number of isomorphism classes in finite
dimensional case.
