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Using a hyperK\"{a}hler rotation on complex structures of a Calabi-Yau 2-foldand rolling of an isotropic 2-submanifold in a symplectic 6-manifold, weconstruct, by gluing, a natural family of immersed Lagrangian deformations of abranched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold andstudy how they deviate from being deformable to a family of special Lagrangiandeformations by examining in detail Joyce's criteria on this family. The resultsuggests a potential image-support rigidity of A-branes that wrap around aspecial Lagrangian 3-sphere in a Calabi-Yau 3-fold, which resembles a similarphenomenon for holomorphic curves that wrap around a rigid smooth rationalcurve in a Calabi-Yau 3-fold in Gromov-Witten theory.

Immersed Lagrangian deformations of a branched covering of a special  Lagrangian 3-sphere in a Calabi-Yau 3-fold and its deviation from Joyce's  criteria: Potential image-support rigidity of A-branes that wrap around a sL  $S^3$