In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of nonlinear Schrödinger–Korteweg-De Vries (NLS–KdV, in short) equations with Hardy potentials: \(\begin{aligned} \left\{ \begin{aligned} (-\Delta )^{s_{1}} u - \lambda _{1} \frac{u}{|x|^{2s_{1}}} - u^{2_{s_{1}}^{*}-1}&= 2\nu h(x) u v &\quad \text{ in } ~ {\mathbb {R}}^{N},\\ (-\Delta )^{s_{2}} v - \lambda _{2} \frac{v}{|x|^{2s_{2}}} - v^{2_{s_{2}}^{*}-1}&= \nu h(x) u^{2}&\quad \text{ in } ~ {\mathbb {R}}^{N},\\ u,v >0 \quad \text{ in } ~ {\mathbb {R}}^{N} \setminus \{0\}, \end{aligned} \right. \end{aligned}\) where \(s_{1},s_{2} \in (0,1)~\text {and}~\lambda _{i}\in (0, \Lambda _{N,s_{i}})\) with \(\Lambda _{N,s_{i}} = 2 \pi ^{N/2} \frac{\Gamma ^{2}(\frac{N+2s_i}{4}) \Gamma (\frac{N+2s_i}{2})}{\Gamma ^{2}(\frac{N-2s_i}{4}) ~|\Gamma (-s_{i})|}, (i=1,2)\) . We obtain ground-state solutions using the concentration-compactness principle and the mountain-pass theorem by imposing certain assumptions on the parameter \(\nu \) and on the function h.
