Karger and Novak [1 ] has shown that the integral curves of a linear vector field X on E3 which has a matrix “ A C~ are: _ 0 1 _ (i) . Helices with common axes and the same parameter if rank [AC] — 3, (ii) . Circles which lie on planes parallel to each other, and which have centers on the axis per- pendicular to those parallel planes, if rank [AC] = 2, (iii) .Parallel straight lines, if rank [AC] = 1. In this study, the results of Karger and Novak are extended to E2n+1 where n > l. It is shown that of ali the results are also valid in this general case and some further elaborations are included.