Summary: We prove the existence of a solution to the Monge-Ampère equation \(\det\text{Hess}(\varphi)=1\) on a cone over a thrice-punctured two-sphere. The total space of the tangent bundle is thereby a Calabi-Yau manifold with flat special Lagrangian fibers. (Each fiber can be quotiented to three-torus if the affine mondrome can be shown to lie in \(\text{SL} (3,\mathbb Z)\ltimes \mathbb R^3\).) Our method is through Baues and Cortés’s result that a metric cone over an elliptic affine sphere has a parabolic affine sphere structure (i.e., has a Monge-Ampère solution). The elliptic affine sphere structure is determined by a semilinear PDE on \(\mathbb C\mathbb P^1\) minus three points, and we prove existence of a solution using the direct method in the calculus of variations.