  From the product of two elliptic curves, Shioda and Inose constructed an
elliptic $K3$ surface having two $\mathrm{II}^*$ fibers. Its Mordell-Weil
lattice structure depends on the morphisms between the two elliptic curves. In
this paper, we give a method of writing down generators of the Mordell-Weil
lattice of such elliptic surfaces when two elliptic curves are $3$-isogenous.
In particular, we obtain a basis of the Mordell-Weil lattice for the singular
$K3$ surfaces $X_{[3,3,3]}$, $X_{[3,2,3]}$ and $X_{[3,0,3]}$.
