An Orthocenter Pins the Ellipse

$F_1, F_2$ are the foci of the ellipse $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$). The point $P\left(\dfrac{2\sqrt6}{3}, 1\right)$ lies on $C$, and the orthocenter of $\triangle F_1PF_2$ is $H\left(\dfrac{2\sqrt6}{3}, -\dfrac53\right)$.

1. Find the equation of $C$. 2. Let $A$ be the left vertex. A line $\ell$ through $F_2$ meets $C$ at $D$ and $E$; the slopes of $AD$ and $AE$ are $k_1, k_2$. If $k_1 + k_2 = -\dfrac12$, find the equation of $\ell$.