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The complete classification of WZNW modular invariant partition functions isknown for very few affine algebras and levels, the most significant being alllevels of $A_1$ and $A_2$ and level 1 of all simple algebras. Here, we addressthe classification problem for the nicest high rank semi-simple affinealgebras: $(A_1^{(1)})^{\oplus_r}$. Among other things, we explicitly find allautomorphism invariants, for all levels $k=(k_1,\ldots,k_r)$, and complete theclassification for $A_1^{(1)}\oplus A_1^{(1)}$, for all levels $k_1,k_2$. Wealso solve the classification problem for $(A_1^{(1)})^{\oplus_r}$, for anylevels $k_i$ with the property that for $i\ne j$ each $gcd(k_i+2,k_j+2)\leq 3$.In addition, we find some physical invariants which seem to be new. Togetherwith some recent work by Stanev, the classification for all$(A^{(1)}_1)^{\oplus_r}_k$ could now be within sight.Comment: 38 pp, (plain tex

Towards a classification of su(2)⊕...⊕su(2) modular invariant partition functions