elliptic fibrations $E(\tilde{\Gamma})/X(\tilde{\Gamma})/{{\mathbb Z}}[1/M]$ will be given in Section 4 below.

[**Remarks**]{}:

*The list in Table 2 exhausts all of the families of semi-stable elliptic $K3$ surfaces with exactly six singular fibers, which correspond to torsion-free genus zero [*congruence*]{} subgroups of $PSL_2({{\mathbb Z}})$ of index $24$. The $6$-tuples of natural numbers appearing in the third column add up to $24$. Therefore, the number of such $6$-tuples is a priori finite. That this list is complete was proved by Sebbar \[Se\].*

\(2) Miranda and Persson \[MP\] studied all possible configurations of $I_n$ fibers on elliptic $K3$ surfaces. In the case of exactly six singular fibers, they obtained $112$ possible configurations including the above nine cases. All these $K3$ surfaces have the maximal possible Picard number $20$. It should be emphasized that the exactly nine configurations correspond to genus zero congruence subgroups of $PSL_2({{\mathbb Z}})$ of index $24$.

\(3) The theory of Miranda and Persson had been extended to prove the uniqueness (over ${{\mathbb C}}$) of K3 surfaces having each of these types of singular fibers by Artal-Bartolo, Tokunaga and Zhang \[BHZ\]. Confer also the article of Shimada and Zhang \[SZ\] for a useful table of extremal elliptic $K3$ surfaces.

The non-rigid 