This paper deals with the Neumann problem for a Keller-Segel system with rotational flux: \(\begin{aligned} \left\{ \begin{array}{ll} {u_{t}=\Delta u-\nabla \cdot (uS_{\theta (x)}\nabla v)}, & x\in \Omega ,\, t>0,\\ {0=\Delta v-v+u}, & x\in \Omega ,\, t>0, \end{array}\right. \end{aligned}\) where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^2\) , \(S_{\theta (x)}=\Big ( \begin{array}{cc} \cos \theta (x) & -\sin \theta (x) \\ \sin \theta (x) & \cos \theta (x) \end{array} \Big )\) is a \(2\times 2\) matrix and the rotation angle satisfies that \(-\frac{\pi }{2}<\theta _0 \leqslant \theta (x) \leqslant \theta _1<\frac{\pi }{2}\) , \(\theta (x)\) is a Lipschitz continuous function. It is shown that for each \(m >\frac{8\pi }{\min \{\cos {\theta _0}, \cos {\theta _1}\}}\) , there exists a classical solution with \(\int _{\Omega }u_0(x)\textrm{d}x=m\) , which blows up at some interior points of \(\Omega \) in finite time; if \(\partial \Omega \) contains a line segment h, then for each \(m>\frac{4\pi }{\min \{\cos {\theta _0}, \cos {\theta _1}\}}\) , there exists a classical solution with \(\int _{\Omega }u_0(x)\textrm{d}x=m\) , which blows up at some points on h in finite time.
