Dimension of orbits of linear time-invariant singular systems under restricted system equivalence

We consider the set of quadruples of matrices defining regular singular linear time-invariant dynamical systems under restricted system equivalence and derive miniversal deformations from a basis of the normal space to orbits under the Lie group action related to this equivalence relation. A lower bound and an upper bound for the dimension of the orbits are obtained. We conclude with examples and further comments about genericity.