X-simple image eigencones of tropical matrices

We investigate max-algebraic (tropical) one-sided systems A⊗x=bA⊗x=b where b is an eigenvector and x lies in an interval X. A matrix A is said to have X-simple image eigencone associated with an eigenvalue λ, if any eigenvector x associated with λ and belonging to the interval X   is the unique solution of the system A⊗y=λxA⊗y=λx in X. We characterize matrices with X-simple image eigencone geometrically and combinatorially, and for some special cases, derive criteria that can be efficiently checked in practice.