In http://mathoverflow.net/a/44885/4177, Tilman points out that one can use a $K3$ surface minus the zeroes of a generic vector field to build a nullcobordism for $24[SU(2)]$. Given that a) this is a purely topological notion and b) we can write down defining equations for smooth affine $K3$s, it may well be easy for experts to write down an explicit and, ideally, highly symmetric vector field on eg the Fermat quartic with 24 zeroes.

More generally: what's an example, algebraic or otherwise, of such a vector field on an explicitly given $K3$ surface?