I'm looking for a few concrete examples of vector bundles on hyperkahler varieties of dimension $\ge 4$. Here are a few examples I know already:

For $X$= the Hilbert scheme of points $S^{[n]}$ on a K3 surface with a line bundle $L$, one has the tautological bundle $L^{[n]}$, which has rank $n$.

For $X$=the variety of lines of a cubic fourfold, one has the restriction of the universal bundles on the ambient Grassmannian. Similarly for other hyperkahler varieties embedded in Grassmannians.

Are there other natural examples of interesting vector bundles like this? (I'm mostly interested in hyperkahler manifolds of $K3^{[n]}$-type, but also other cases are interesting).