  We study the geometry of M5-branes wrapping a 2-cycle which is Special
Lagrangian with respect to a specific complex structure in a Calabi-Yau
two-fold. Using methods recently applied to the three-fold case, we are again
able find a characterization of the geometry, in terms of a non-integrable
almost complex structure and a (2,0) form. This time, however, due to the
hyper-K{\"a}hler nature of the underlying 2-fold we also have the freedom of
choosing a different almost complex structure with respect to which the wrapped
2-cycle is holomorphic. We show that this latter almost complex structure is
integrable. We then relate our geometry to previously found geometries of
M5-branes wrapping holomophic cycles and go further to prove some previously
unknown results for M5-branes on holomorphic cycles.
