Kinematic similarity of linear differential systems with a parametric coefficient of a derivative

The kinematic similarity set for a pair of two piecewise continuous bounded matrices (i.e., matrix-valued functions) $A(\cdot)$ and $B(\cdot)$ defined on the time semi-axis is defined as the set of all real $\mu$ for which the matrices $\mu A(\cdot)$ and $\mu B(\cdot)$ are kinematically similar. It is shown that a subset of the real axis is a kinematic similarity set for a pair of matrices if it contains the origin and is an $F\sigma$-set.