On a theorem of Picard

We extend Picard's theorem on the existence of elliptic solutions of the second kind of linear homogeneous nth_order scalar ordinary differential equations with coefficients being elliptic functions (associated with a common period lattice) to linear homogeneous first-order n x n systems. In particular, the qualitative Floquet-type structure of fundamental systems of solutions in terms of elliptic and exponential functions, polynomials, and Weierstrass zeta functions of the independent variable is determinod. Connections with completely integrable systems are mentioned. In order to set the stage for our extension of Picard's theorem on the existence of elliptic solutions of the second kind of nth_order scalar ordinary differential equations with elliptic coefficients (with a common period lattice) to first-order n x n systems and an explicit description of the corresponding Floquet-type structure of fundamental systems of solutions, we briefly review Floquet theory for singly periodic n x n systems. Denote by M(n), n E N, the set of n x n matrices with entries in C, define GL(n) {A E M(n) I det(A) 7 0}, and consider the linear homogeneous system (1) @'I(z) Q(z)'J'(z), z E C, where @(z) E GL(n), Q(z) E M(n), with Q(z) a continuous periodic matrix of period Q E C\{0}, that is, (2) Q(z+Q) = Q(z), z E C. Concerning (continuously differentiable) fundamental matrices 4>(z) of solutions of (1), one has the following basic Floquet theorem (see, e.g., [5], Ch. 3, [20], Ch. 4, [30], Ch. 5). Theorem 1. Let Q(z) E M(n) with Q(z) a continuous periodic matrix of period Q E C\{0}. Then (1) admits a fundamental matrix 4?(z) E GL(n) of the type (3) 4(z) = P(z) exp (zK), z E C, where P(z) E GL(n), K E M(n), and P(z) is continuously differentiable and periodic of period Q, (4) P(z+Q) = P(z), z E C. Received by the editors September 23, 1996. 1991 Mathematics Subject Classification. Primary 33E05, 34C25; Secondary 58F07. The research was based upon work supported by the National Science Foundation under Grant No. DMS-9623121. (g)1998 American Mathematical Society