Stationary homogeneous solutions for the inviscid SQG equation

In this paper, we prove existence of stationary nontrivial homogeneous solutions for SQG equation with infinite energy (unbounded at the infinity). Our analysis also covers the existence of stationary solutions for generalized De Gregorio equation $w_t+\alpha u{w}_x=u_{x}w,u_x=Hw$ with $\alpha >\frac{1}{2}\mathrm{.}$