Take a line $L$ of the first type on a smooth cubic threefold $X$ over $\mathbb C$, then its normal bundle $N_{L|X}$ is isomorphic to $\mathcal{O}_L\oplus \mathcal{O}_L$. This is equivalent to say that there is a $\mathbb P^1$-family of quadric surfaces in $\mathbb P^4$ tangent to $X$ along $L$. I'm trying to write down these quadric surfaces explicitly.

Let $L=\{x_2=x_3=x_4=0\}$, then up to change of coordinates, $X$ has equation

$$x_2x_0^2+x_3x_0x_1+x_4x_1^2+\text{higher order terms in }x_2,x_3,x_4.$$

The dual map at $p={(x_0,x_1)}\in L$ is $\mathcal{D}(p)=[0,0,x_0^2,x_0x_1,x_1^2]$, which determines the hyperplane $T_{p}X$ at $p$ and we just need to find quadric surfaces containing $L$ and have tangent planes at each $p\in L$ contained in $T_{p}X$.

I can find two of such quadric surfaces:

$$x_4=0,~x_2x_0+x_3x_1=0,$$ $$x_2=0,~x_4x_1+x_3x_0=0.$$

Unfortunately, the family is not a linear combination of them and I cannot find any more such quadric surface. Note that in the 1972 paper *The intermediate Jacobian of the cubic threefolds* by Clemens and Griffiths, page 309, some constructions are given in terms of equations of varieties of lines of the quadric surfaces in the Grassmannian $Gr(2,5)$. However, there seems to be a typo in the defining equations (of curve $B(\alpha_0,\alpha_1)$ in the paper), which I couldn't fix.

How to find the entire $\mathbb P^1$-family (hopefully in equations)? Any comments or suggestions will be appreciated!