Positive orthant scalar controllability of bilinear systems

For the bilinear control system $\dot x=(A+uB)x$, $x\in\mathbb R^n$, $u\in\mathbb R$ where $A$ is an $n\times n$ essentially nonnegative matrix, and $B$ is a diagonal matrix, the following controllability problem is investigated: can any two points with positive coordinates be joined by a trajectory of the system? For $n>2$, the answer is negative in the generic case: hypersurfaces in $\mathbb R^n$ are constructed that are intersected by all the trajectories of the system in one direction.