A Constant Slope Combination

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has upper vertex $A(0,1)$ and eccentricity $\dfrac{\sqrt 3}2$.

1. Find the equation of the ellipse $C$. 2. Let $B$ be the lower vertex of $C$, and let $M$ be a moving point at distance $1$ from the origin $O$ (with $M$ distinct from $A$ and $B$). The line $AM$ meets $C$ again at a point $N$. Let $k_1$, $k_2$ be the slopes of $BM$, $BN$. Does there exist a constant $\lambda$ such that $k_1+\lambda k_2=0$ always holds? If so, find $\lambda$; if not, explain why.