In reply to a problem posed by Jean Leray in 1950, a nontrivial example of application of the Nielsen fixed-point theory to differential systems is given. So the existence of two entirely bounded solutions or three periodic (harmonic) solutions of a planar system of ODEs is proved by means of the Nielsen number. Subsequently, in view of T. Matsuoka's results in Invent. Math. (70 (1983), 319-340) and Japan J. Appl. Math. (1 (1984), no. 2, 417434), infinitely many subharmonics can be generically deduced for a smooth system. Unlike in other papers on this topic, no parameters are involved and no simple alternative approach can be used for the same goal. 1. HISTORICAL REMARKS Unlike any standard fixed-point principle, Nielsen theory gives us additional information about the lower estimate of the fixed points number. It was originated by the Danish mathematician Jakob Nielsen in 1927 [N], who studied with this respect self-maps of compact surfaces. Later on, finite polyhedra were systematically treated by FRanz Wecken in 1941-42 [W], who also gave an alternative definition of the Nielsen relations which we use below. The crucial step in this development was also accomplished by Andrzej Granas who defined a strictly related fixed-point index for arbitrary (i.e. also infinite-dimensional) ANRs in 1972 [G]. Thus, a sufficiently general fixed-point principle, preserving the number of essential classes of fixed points under homotopy, could be formulated by U. K. Scholz [S], Boju Jiang [Bol], R. F. Brown [Brl], and some others (see the references in [AGJ1], [Bol], [Brl]). By sufficiency, we mean the appropriate form which was suitable for answering the question posed already by Jean Leray [L] at the first International Congress of Mathematicians held after World War II in Cambridge, Mass., in 1950. Namely, he suggested the problem of adapting the Nielsen theory to the needs of nonlinear analysis and, in particular, of its application to differential systems for obtaining multiplicity results (for more details see e.g. [Br3], [Br4]). Since that time only several papers have been devoted to this problem (see e.g. [BKM], [Br2], [Br3], [Br4], [Fl], [F2]), but either additional parameters had to be Received by the editors May 4, 1998 and, in revised form, November 6, 1998. 1991 Mathematics Subject Classification. Primary 34B15, 47H10.