In Eq. (M), P, Q, X are real ?z x n matrices, and in (V), Q, x are real nvectors and g is real valued. A real n > 0, constant) is said to be “oscillatory ” if its determinant det X(t) has an infinity of zeros on [ar, m). Otherwise X(t) is called “nonoscillatory.” A real n-vector x(t), t E [01, cc) (a > 0, constant), is said to be “oscillatory” if (x(t), h) has an infinity of zeros on [c+ co) for any h E R” (R = (-CO, co)). Here (., .) denotes the inner product of R”. Although there is a large number of oscillation criteria for solutions of (M) with Q(t) = 0, problem (NI) with Q(t) + 0 becomes extremely difficult, and there are virtually no results concerning the oscillation of (RI) even in simple linear cases. One of the main reasons for this is the absence of “prepared” solutions of (NI). Prepared solutions of (M) with Q(t) = 0 are certain solutions with a symmetry property that can be effectively employed in oscillation theory. Our main purpose here is to give a result for (RI), where P satisfies an integral condition involving a second antiderivative It(t) of Q, and F’(t) is bounded on [01, 00) (with ci > 0, constant) and “stays away from zero,” in a sense to be more precise in Corollary 1. This result extends to the present case a result of Kartsatos and Manougian (see [4]) concerning Izth-order scalar equations. As far as (V) is concerned, we first give an oscillation result with Q(t, .r) = Q(t) which extends a second-order result of Domshlak [l] for Q(t) G 0. Our method is based on reducing the problem to a scalar equation, and then applying a comparison result of Kartsatos [3]. L% ‘e also give a second result, which deals