Reducibility of the Moduli Space of Stable Rank $2$ Reflexive Sheaves with Chern Classes $c_1=-1$, $c_2=4$, $c_3=2$ on Projective Space $\mathbb{P}^3$

We prove the reducibility of the moduli space $M_{\mathbb{P}^3}^{\mathrm{ref}}(2;-1,4,2)$ of stable rank 2 reflexive sheaves with Chern classes $c_1=-1$, $c_2=4$, $c_3=2$ on projective space $\mathbb {P}^3$. This gives the first example of a reducible space in the series of moduli spaces of stable rank 2 reflexive sheaves with Chern classes $c_1=-1$, $c_2=4$, $c_3=2m$, $m=1,2,3,4,5,6,8$. We find two components of the expected dimension 27 of this space and give their geometric description via the Serre construction.