Special Lagrangians with Cylindrical Tangent Cones

We construct new examples of special Lagrangian submanifolds $Y\subset \mathbf{C}^{n+1}$, $n\geq 3$ 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.