Let $S$ be a smooth projective surface with an ample divisor $X\subset S$. Consider the moduli stack of vector bundles $F$ on $S$ such that

1) $c_1(F)=0$

2) $c_2(F)=n$

3) The restriction of $F$ to $X$ is fixed (say, isomorphic to a bundle $M$).

$\mathbf{Questions:}$

1) Is it true that for any $n$ and $M$ the above stack is of finite type?

2) Is it true, that for any given $M$ the above stack is empty for $n$ sufficiently small?