Example - Flow $(\varphi_t)_{t\in\mathbb{R}}$ ergodic, but not each $\varphi_t$

Question

Let $(X,\mathfrak{B},\mu)$ be a probability space and $(\varphi_t)_{t\in\mathbb{R}}$ a flow, i.e. $\varphi_t:X\rightarrow X$ is measure-preserving and $\varphi_0(x)=x$ as well as $\varphi_{s+t} (x)=\varphi_s\circ\varphi_t(x)$ are satisfied for all $x\in X, ~s,t\in\mathbb{R}$. We call this flow ergodic if $$\bigg(\mu\big(B\triangle \varphi^{-1}_t(B)\big)=0 ~~\forall t\in\mathbb{R}\bigg)\Rightarrow \mu(B)\in\{0,1\}.$$ What is an example for a flow that is ergodic but some of the systems $(X,\mu,\varphi_t)$ are not?