Trisecting a Chord

The hyperbola $E \colon \dfrac{x^2}{4} - y^2 = 1$ meets the line $l \colon y = kx - 3$ at two points $A$ and $B$, and $M$ is the midpoint of segment $AB$.

1. As $k$ varies, find the equation of the locus of $M$. 2. Suppose $l$ also meets the two asymptotes of $E$ at points $C$ and $D$. Does there exist a real number $k$ such that $A$ and $B$ are the two trisection points of segment $CD$? If so, find $k$; if not, explain why.