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Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvableBaumslag-Solitar group for $n \geq 2$. We study representations of $BS(1, n)$on the plane by orientation preserving homeomorphisms, assuming that $a$ actsas a linear map and $b$ as a map with bounded displacement. We find that thepossibilities for a faithful action depend greatly on the Jordan canonical formof the map $h$ defined by the action of $a$. In case $h$ is diagonalizable over$\mathbb R$, we shall give examples or prove rigidity theorems depending on theeigenvalues. We also show some rigidity in the cases where $h$ is elliptic orparabolic. Then we give applications to the actions of $BS(1, n)$ byhomeomorphisms of the torus.

One some planar Baumslag-Solitar actions