ON NUMBER AND EVENNESS OF SOLUTIONS OF THE SU(3) TODA SYSTEM ON FLAT TORI WITH NON-CRITICAL PARAMETERS

We study the SU(3) Toda system with singular sources{Delta u + 2e(u) - e(v) = 4 pi Sigma(m)(k=0) n(1,k)delta(pk) on E tau,Delta v + 2e(v) - e(u) = 4 pi Sigma(m)(k=0) n(2,k)delta(pk) on E tau,where E-tau := C/(Z + Z tau) with Im tau > 0 is a flat torus, delta(pk) is the Dirac measure at p(k), and n(i,k) is an element of Z(>= 0) satisfy Sigma(k) n(1,k) (sic) Sigma(k) n(2,k) mod 3. This is known as the non-critical case and it follows from a general existence result of [3] that solutions always exist. In this paper we prove that(i) The system has at most1/3 x 2(m+1) pi(k=0) (m) (n(1,k) + 1)(n(2,k) + 1)(n(1,k) + n(2,k )+ 2) is an element of Nsolutions. We have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical Bezout theorem from algebraic geometry.(ii) For m = 0 and p0 = 0, the system has even solutions if and only if at least one of {n(1,0), n(2,0)} is even. Furthermore, if n(1,0) is odd, n(2,0) is even and n(1,0) < n(2,0), then except for finitely many tau's modulo SL(2, Z) action, the system has exactly n(1,0)+1/2 even solutions.Differently from [3], our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.