The paper studies aspects of self-similarity for special flows on surfaces. More precisely, the author considers smooth measure-preserving flows on compact connected orientable surfaces of genus at least \(2\) with a finite number of non-degenerate singular points and no saddle connections. It is shown that in this class there exist flows with no self-similarities, that is, flows \(\{T_t\}_{t\in {\mathbb R}}\) such that \(\{T_t\}_{t\in {\mathbb R}}\) and \(\{T_{st}\}_{t\in {\mathbb R}}\) are not measure-theoretical isomorphic, for any \(s \in {\mathbb R} \setminus\{-1,1\}\). This settles an open problem which was raised by K. Frączek and M. Lemańczyk [Proc. Lond. Math. Soc. (3) 99, No. 3, 658–696 (2009; Zbl 1186.37008)].
The idea of the proof is to build special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities. For these flows, it is shown that they are neither partially rigid nor spectral self-similar, which then allows to deduce the desired non-self-similarity property.