A Tangent Aligned with a Chord

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has right focus $F$ and upper vertex $B$, with eccentricity $\dfrac{2\sqrt 5}5$ and $|BF|=\sqrt 5$.

1. Find the equation of the ellipse. 2. A line $l$ has exactly one common point $M$ with the ellipse and meets the positive $y$-axis at a point $N$. The line through $N$ perpendicular to $BF$ meets the $x$-axis at a point $P$. If $MP\parallel BF$, find the equation of $l$.