On a Theorem of Picard

We extend Picard's theorem on the existence of elliptic solutions of the second kind of linear homogeneous n th-order scalar ordinary diierential equations with coeecients being elliptic functions (associated with a common period lattice) to linear homogeneous rst-order n n systems. In particular, the qualitative structure of the general solution in terms of elliptic and exponential functions, polynomials, and Weierstrass zeta functions of the independent variable is determined. Connections with completely integrable systems are also 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 n th-order scalar ordinary diierential equations with elliptic coeecients (with a common period lattice) to rst-order n n systems, we brieey review Floquet theory for singly periodic n n systems. (Due to our interest in doubly periodic coeecients later on we immediately use a formulation in terms of complex variables rather than real variables.) Denote by M(n), n 2 N the set of n n matrices with entries in C , deene GL(n) := fA 2 M(n) j det(A) 6 = 0g, and consider the linear homogeneous system 0 (z) = Q(z))(z); z 2 C ; (1) where () 2 GL(n), Q() 2 M(n), with Q() a continuous periodic matrix of period 2 C nf0g, that is, Q(z +) = Q(z); z 2 C : (2) Concerning (continuously diierentiable) fundamental matrices (z) of solutions of (1), one has the following basic Floquet theorem (see, e.