Closed irregularity sets of linear differential systems with a parameter multiplying the derivative

For any closed subset M of the real line that does not contain zero, we construct a linear differential system with bounded piecewise continuous coefficient matrix A(·) such that the corresponding system with coefficient matrix μA(·) linearly depending on a real parameter μ is Lyapunov irregular for all μ in M and Lyapunov regular for all other parameter values.