Closed-form solutions are derived for two problems of nonlinear elastic fracture mechanics. The cases considered deal with the out-of-plane deformation of a centrally-cracked cylinder of elliptic cross section involving hyperelastic materials of either Neo-Hookean or Mooney-Rivlin type. Each solid is loaded by a self-equilibrating anti-plane shear traction applied to the faces of its crack and has its remote boundary either free or fixed. It is shown that, as in the case of small strains, the equation governing the out-of-plane response decouples from those defining in-plane behavior. It is also found that a finite state of pure out-of-plane deformation can only be sustained in the presence of Poynting stresses. In particular, nonlinear solids are seen to require out-of-plane direct axial stresses to satisfy the prescribed kinematics of deformation. Additionally, the non-regular shear stresses at the crack tips retain the same power of singularity as would exist in their linear elastic counterparts. Interestingly enough, however, the direct stresses, which are regular in linear materials, exhibit a singularity of higher order than that of the shear stresses.