Given are tuples $ (a_{11},\dots,a_{1k}), (a_{21},\dots,a_{2k}), \dots, (a_{n1},\dots,a_{nk})$ . We want to know if there is a partition of the tuples into two parts, so that for every coordinate $ i=1,\dots,k$ , the sum of each part is equal to the sum of the other part.

Has this problem been studied before? If $ k=1$ , it is of course the famous Partition problem. But I cannot find this generalization under the section “Variants and generalizations” in the Wikipedia page.