Equal Triangle Areas on a Conic

In the coordinate plane $xOy$, let $B$ be the reflection of $A(-1,1)$ across the origin, and let $P$ be a moving point such that the product of the slopes of $AP$ and $BP$ is $-\dfrac 13$.

1. Find the equation of the locus of $P$. 2. The lines $AP$ and $BP$ meet the line $x=3$ at points $M$ and $N$ respectively. Does there exist a point $P$ for which $\triangle PAB$ and $\triangle PMN$ have equal areas? If so, find its coordinates; if not, explain why.