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Mukai’s program in [16] seeks to recover a K3 surface X from any curve C on it by exhibiting it as a Fourier–Mukai partner to a Brill–Noether locus of vector bundles on the curve. In the case X has Picard number one and the curve 
$C\in |H|$ is primitive, this was confirmed by Feyzbakhsh in [11, 13] for 
$g\geq 11$ and 
$g\neq 12$. More recently, Feyzbakhsh has shown in [12] that certain moduli spaces of stable bundles on X are isomorphic to the Brill–Noether locus of curves in 
$|H|$ if g is sufficiently large. In this paper, we work with irreducible curves in a nonprimitive ample linear system 
$|mH|$ and prove that Mukai’s program is valid for any irreducible curve when 
$g\neq 2$, 
$mg\geq 11$ and 
$mg\neq 12$. Furthermore, we introduce the destabilising regions to improve Feyzbakhsh’s analysis in [12]. We show that there are hyper-Kähler varieties as Brill–Noether loci of curves in every dimension.
