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We partially extend to hyperk\"ahler fourfolds of Kummer type the resultsthat we have proved regarding stable rigid vector bundles on hyperk\"ahler (HK)varieties of type $K3^{[n]}$. Let $(M,h)$ be a general polarized HK fourfold ofKummer type such that $q_M(h)\equiv -6\pmod{16}$ and the divisibility of $h$ is$2$, or $q_M(h)\equiv -6\pmod{144}$ and the divisibility of $h$ is $6$. We showthat there exists a unique (up to isomorphism) slope stable vector bundle $\calF$ on $M$ such that $r({\cal F})=4$, $ c_1({\cal F})=h$, $\Delta({\calF})=c_2(M)$. Moreover $\cal F$ is rigid. One of our motivations is the desireto describe explicitly a locally complete family of polarized HK fourfolds ofKummer type.

Rank 4 stable vector bundles on hyperk\"ahler fourfolds of Kummer type