  Let $X$ be a projective K3 surface with generic polarization $\cO_X(1)$ and
let $M_c=M(2,0,c)$ be the moduli space of semistable torsion-free sheaves on
$X$ of rank 2, with Chern classes $c_1=0$ and $c_2=c$. When $c=2n\ge 4$ is
even, $M_c$ is a singular projective variety. We show that there is no
symplectic desingularization of $M_{2n}$ if $\frac{n a_n}{2n-3}$ is not an
integer where $a_n$ is the Euler number of the Hilbert scheme $X^{[n]}$ of $n$
points in $X$.
