This paper is concerned with the system of ordinary linear differential equationswithA(Z) (n:)-matrix of meromorphic functions in a neighborhood of the (isolated) singular point 0 anda column vector of dimensionn. First we prove a new theorem about the reduction of such systems, by which a generally known result of J. Moser [12] from 1960 is essentially improved. With the help of a method from H. L. Turrittin [13], which was extended by us, this result finally leads to a lower bound for the reduction. Then according to our former result [7] this is also a lower bound for the maximum order ρmaxof a fundamental solution matrix: ρmax≥(1/n)δ(A). The quantity δ(A) is the limit of a sequence of quantities which characterize the singular behavior inz= 0. We will show that δ(A) is an invariant and that δ(A) = 0 holds if and only if the system is regular singular inz= 0. At the end of the paper the results will be applied to ordinary linear differential equations.