  Let $X$ be a smooth projective curve with genus $g\geq3$. Let $\mathcal{N}$
be the moduli space of stable rank two vector bundles on $X$ with a fixed
determinant $\mathcal{O}_X(-x)$ for $x\in X$. In this paper, as a
generalization of Kiem and Castravet's works, we study the stable maps in
$\mathcal{N}$ with genus $0$ and degree $3$. Let $P$ be a natural closed
subvariety of $\mathcal{N}$ which parametrizes stable vector bundles with a
fixed subbundle $L^{-1}(-x)$ for a line bundle $L$ on $X$. We describe the
stable map space $\mathbf{M}_0(P,3)$. It turns out that the space
$\mathbf{M}_0(P,3)$ consists of two irreducible components. One of them
parameterizes smooth rational cubic curves and the other parameterizes the
union of line and smooth conics.
