  The present paper is another step toward the classification of
$\Z\times\Z$-graded Lie algebras: we classify $\Z\times\Z$-graded Lie algebras
$A=\oplus_{i,j\in\Z}A_{i,j}$ over a field $F$ of characteristic 0 with dim
$A_{i,j}\le1$ for $i,j\in\Z$, satisfying: (1) $\AA_0=\oplus_{i\in \Z}\AA_{i,0}$
is isomorphic to the centerless Virasoro algebra, (2)
$A_{0,-1},A_{0,0},A_{0,1}$ span the 3-dimensional simple Lie algebra $sl_2$,
(3) $A$ is generated by the centerless Virasoro algebra and $sl_2$. These
algebras include some rank 2 centerless Virasoro algebras, and some rank 2
Block algebras and their extensions.
