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From the product of two elliptic curves, Shioda and Inose constructed anelliptic $K3$ surface having two $\mathrm{II}^*$ fibers. Its Mordell-Weillattice structure depends on the morphisms between the two elliptic curves. Inthis paper, we give a method of writing down generators of the Mordell-Weillattice 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 $X_{[3,0,3]}$.

Mordell-Weil lattice of Inose's Elliptic $K3$ surface arising from the product of 3-isogenous elliptic curves