  Microswimmers (planktonic microorganisms or artificial active particles)
immersed in a fluid interact with the ambient flow, altering their
trajectories. By modelling anisotropic microswimmers as spheroidal bodies with
an intrinsic swimming velocity that supplements advection and reorientation by
the flow, we investigate how shape and swimming affect the trajectories of
microswimmers in surface gravity waves. The coupling between flow-induced
reorientations and swimming introduces a shape dependency to the vertical
transport. We show that each trajectory is bounded by critical planes in the
position-orientation phase space that depend only on the shape. We also give
explicit solutions to these trajectories and determine whether microswimmers
that begin within the water column eventually hit the free surface. We find
that it is possible for microswimmers to be initially swimming downwards, but
to recover and head back to the surface. For microswimmers that are initially
randomly oriented, the fraction that hit the free surface is a strong function
of shape and starting depth, and a weak function of swimming speed.
