  Using a hyperK\"{a}hler rotation on complex structures of a Calabi-Yau 2-fold
and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we
construct, by gluing, a natural family of immersed Lagrangian deformations of a
branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and
study how they deviate from being deformable to a family of special Lagrangian
deformations by examining in detail Joyce's criteria on this family. The result
suggests a potential image-support rigidity of A-branes that wrap around a
special Lagrangian 3-sphere in a Calabi-Yau 3-fold, which resembles a similar
phenomenon for holomorphic curves that wrap around a rigid smooth rational
curve in a Calabi-Yau 3-fold in Gromov-Witten theory.
