  We investigate max-algebraic (tropical) one-sided systems $A\otimes x=b$
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
$\lambda$, if any eigenvector $x$ associated with $\lambda$ and belonging to
the interval $X$ is the unique solution of the system $A\otimes y=\lambda x$ 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.
