  Planar linear flows are a one-parameter family, with the parameter
$\hat{\alpha}\in [-1,1]$ being a measure of the relative magnitudes of
extension and vorticity; $\hat{\alpha} = -1$, $0$ and $1$ correspond to
solid-body rotation, simple shear flow and planar extension, respectively. For
a neutrally buoyant spherical drop in a hyperbolic planar linear flow with
$\hat{\alpha}\in(0,1]$, the near-field streamlines are closed for $0 \leq
\hat{\alpha} < 1$ and for $\lambda > \lambda_c = 2 \hat{\alpha} / (1 -
\hat{\alpha})$, $\lambda$ being the drop-to-medium viscosity ratio; all
streamlines are closed for an ambient elliptic linear flow with
$\hat{\alpha}\in[-1,0)$. We use both analytical and numerical tools to show
that drop deformation, as characterized by a non-zero capillary number ($Ca$),
destroys the aforementioned closed-streamline topology. While inertia has
previously been shown to transform closed Stokesian streamlines into open
spiraling ones that run from upstream to downstream infinity, the streamline
topology around a deformed drop, for small but finite $Ca$, is more
complicated. Only a subset of the original closed streamlines transforms to
open spiraling ones, while the remaining ones densely wind around a
configuration of nested invariant tori. Our results contradict previous efforts
pointing to the persistence of the closed streamline topology exterior to a
deformed drop and have important implications for transport and mixing.
