Flows with uncountable but meager group of self-similarities

Given an ergodic probability preserving flow $T=(T_t)_{t\in\Bbb R}$, let $I(T):=\{s\in\Bbb R^*\mid T\text{is isomorphic to}(T_{st})_{t\in\Bbb R}\}$. A weakly mixing Gaussian flow $T$ is constructed such that $I(T)$ is uncountable and meager. For a Poisson flow $T$, a subgroup $I_{\text{Po}}(T)\subset I(T)$ of Poissonian self-similarities is introduced. Given a probability measure $\kappa$ on $\Bbb R^*_+$, a zero-entropy Poisson flow $T$ is constructed such that $I_{\text{Po}}(T)$ is the group of $\kappa$-quasi-invariance.