We consider the existence problem of the following singular Toda system on a compact Riemann surface \((\Sigma , g)\) without boundary \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _gu_1=2\overline{\rho }_1\Big ({\frac{h_1e^{u_1}}{\int _{\Sigma }h_1e^{u_1}dV_g}}-1\Big )-\rho _2\Big ({\frac{h_2e^{u_2}}{\int _{\Sigma }h_2e^{u_2}dV_g}}-1\Big )-4\pi \alpha _1(\delta _0-1),\\ \\ -\Delta _gu_2=2\rho _2\big ({\frac{h_2e^{u_2}}{\int _{\Sigma }h_2e^{u_2}dV_g}}-1\big )-\overline{\rho }_1\big ({\frac{h_1e^{u_1}}{\int _{\Sigma }h_1e^{u_1}dV_g}}-1\big )-4\pi \alpha _2(\delta _0-1), \end{array}\right. } \end{aligned}\) where \(h_1,\,h_2\) are smooth sign-changing functions, \(\overline{\rho }_1:=4\pi (1+\overline{\alpha }_1),\,0<\rho _2<4\pi (1+\overline{\alpha }_2)\) , and \(\overline{\alpha }_i=\min \{0,\alpha _i\}\) with \(\alpha _i>-1\) for \(i=1,2\) . Relying on the proof framework developed in [14], combined with the Pohozaev identity and classical blow-up analysis, we establish existence results under suitable assumptions. Our results generalize the results of Jost and Wang [18] from regular Toda system with positive weight functions to singular Toda system involving sign-changing weights.
