  We identify incompressible planar linear flows that are generalizations of
the well known one-parameter family characterized by the ratio of in-plane
extension to (out-of-plane) vorticity. The latter `canonical' family is
classified into elliptic and hyperbolic linear flows with closed and open
streamlines, respectively, corresponding to the extension-to-vorticity ratio
being less or greater than unity; unity being the marginal case of simple shear
flow. The novel flows possess an out-of-plane extension, but the streamlines
may nevertheless be closed or open, allowing for an organization, in a
three-dimensional parameter space, into regions of `eccentric' elliptic and
hyperbolic flows, separated by a surface of degenerate linear flows with
parabolic streamlines that are generalizations of simple shear. We discuss
implications for various fluid mechanical scenarios.
