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We consider an unbounded, homogeneous, pre-stressed orthotropic elastic composite containing an elliptical hole and subject to uniform remote tensile and uniform remote tangential shear loads (Mode I and respectively Mode II of fracture). Using the conformal mapping technique and the representation of the stress and displacement fields by complex potentials, we determine the solution of the problem in a compact and elementary form. When the smaller semiaxis of the elliptical hole tends to zero, i.e. the hole becomes a crack, the potentials obtained reduce to a form similar to that of the crack problem, obtained by solving the corresponding Riemann-Hilbert problem.