Exact solutions for static spherically symmetric spacetime with a perfect fluid in Rastall theory

In general relativity, exact Liouvillian solutions for a static and spherically symmetric spacetime with a perfect fluid and the equation of state $p(r)=w\rho (r)$ are known only for $w=0,-\frac{1}{6}, -\frac{1}{5}, -\frac{1}{3}, -1$. We extend this setup to Rastall's theory, presenting the relation between the Rastall parameter and the constant $w$, and deriving exact solutions that correspond to the known counterparts in general relativity, except for $w=-\frac{1}{6}$. Furthermore, we find that, when $w\neq \frac{1}{3}, -1$, there exist several types of solutions whose behavior changes depending on the choice of constants.