Here is the construction.

Start with $U$ a normal variety of dimension 3 with a unique singular point locally isomorphic to the quotient of $(x, y, z) \to (-x, -y, -z)$. Also assume we have a small contraction which contracts a smooth rational curve $C$ through the singular point.

It is known that we can blow up the singular point in $U$ and get a smooth variety $V$ whose exceptional divisor is $P^2$ with normal bundle $\mathcal{O}(-2)$. Now my question is what is the normal bundle of the strict transform of $C$ in $V$?

It is easy to see the intersection number with $K_V$ should be $0$. In the book Geometry of Higer dimensional Algebraic Variety by Miyaoka and Peternell, the author claims it is easy to see the normal bundle is $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ (p.184 Example 7.10). But I do not know why.