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 \begin{equation*} \begin{cases} \Delta u+2e^{u}-e^{v}=4\pi \sum _{k=0}^{m} n_{1,k}\delta _{p_{k}} \quad \text{ on }\; E_{\tau}, \\ \Delta v+2e^{v}-e^{u}=4\pi \sum _{k=0}^{m} n_{2,k}\delta _{p_{k}} \quad \text{ on }\; E_{\tau}, \end{cases} \end{equation*} where $E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$ with $\operatorname{Im}\tau \gt 0$ is a flat torus, $\delta _{p_{k}}$ is the Dirac measure at $p_{k}$, and $n_{i,k}\in \mathbb{Z}_{\geq 0}$ satisfy $\sum _{k}n_{1,k}\not \equiv \sum _{k} n_{2,k} \mod 3$. This is known as the non-critical case and it follows from a general existence result of [$\href{ https://doi.org/10.1016/j.aim.2015.07.036}{3}$] that solutions always exist. In this paper we prove that (i) The system has at most \begin{equation*} \frac{1}{3\times 2^{m+1}}\prod _{k=0}^{m}(n_{1,k}+1)(n_{2,k}+1)(n_{1,k}+n_{2,k}+2) \in \mathbb{N} \end{equation*} solutions. 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 Bézout theorem from algebraic geometry. (ii) For $m=0$ and $p_{0}=0$, 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 $n_{1,0}\lt n_{2,0}$, then except for finitely many $\tau $’s modulo $SL(2,\mathbb{Z})$ action, the system has exactly $\frac{n_{1,0}+1}{2}$ even solutions. Differently from [$\href{ https://doi.org/10.1016/j.aim.2015.07.036}{3}$], 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.