Special Lagrangians with Cylindrical Tangent Cones

We construct new examples of special Lagrangian submanifolds $Y\subset {\mathbf{C}}^{n+1}$, $n\ge 3$ in a neighborhood of the origin, with an isolated singularity, but with cylindrical tangent cone $C\times \mathbf{R}$. Moreover, $Y\setminus \{0\}$ is connected while $(C\setminus \{0\})\times \mathbf{R}$ is not. Such examples exist, for example, when $C$ is a pair of transverse planes.