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We study the $SU(3)$ Toda system with singular sources \[ \begin{cases}\Delta u+2e^{u}-e^v=4\pi\sum_{k=0}^m n_{1,k}\delta_{p_k}\quad\text{ on }\;E_{\tau},\\ \Delta v+2e^{v}-e^u=4\pi \sum_{k=0}^mn_{2,k}\delta_{p_k}\quad\text{ on }\; E_{\tau}, \end{cases} \] where$E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ with$\operatorname{Im}\tau>0$ is a flat torus, $\delta_{p_k}$ is the Dirac measureat $p_k$, 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 thenon-critical case and it follows from a general existence result of \cite{BJMR}that solutions always exist. In this paper we prove that  (i) The system has at most \[\frac{1}{3\times2^{m+1}}\prod_{k=0}^m(n_{1,k}+1)(n_{2,k}+1)(n_{1,k}+n_{2,k}+2)\in\mathbb{N}\]solutions. We have several examples to indicate that this upper bound should besharp. Our proof presents a nice combination of the apriori estimates fromanalysis and the classical B\'{e}zout theorem from algebraic geometry.  (ii) For $m=0$ and $p_0=0$, the system has even solutions if and only if atleast 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 $\tau$'smodulo $SL(2,\mathbb{Z})$ action, the system has exactly $\frac{n_{1,0}+1}{2}$even solutions.  Differently from \cite{BJMR}, our proofs are based on the integrability ofthe Toda system, and also imply a general non-existence result for evensolutions of the Toda system with four singular sources.

On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters