We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over
$\mathbb{C}$
using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on
$\mathbb{N}$
-graded Lie algebras of maximal class. As shown by
$\text{A}$
. Fialowski there are only three isomorphism types of
$\mathbb{N}$
-graded Lie algebras
$L\,=\,\oplus _{i=1}^{\infty }\,{{L}_{i}}$
of maximal class generated by
${{L}_{i}}$
and
${{L}_{2}}$
,
$L\,=\,\left\langle {{L}_{1}},\,{{L}_{2}} \right\rangle$
. Vergne described the structure of these algebras with the property
$L\,=\,\left\langle {{L}_{1}} \right\rangle$
. In this paper we study those generated by the first and
$q$
-th components where
$q\,>\,2$
,
$L\,=\,\left\langle {{L}_{1}},\,{{L}_{q}} \right\rangle$
. Under some technical condition, there can only be one isomorphism type of such algebras. For
$q=\,3$
we fully classify them. This gives a partial answer to a question posed by Millionshchikov.