Consider for instance the case of $M:=\mathbb{R}$, and any $C^k$ field $X$ on $M$, that we may identify with a function. Say, such that $0 < a \le X(s)\le b$. The field $X$ has a globally defined flow, conjugated to the translation flow $x\mapsto x+t$: $$\phi^t(x):= h^{-1}(h(x)+t)\\ ,$$ where the conjugation $h:\mathbb{R}\to\mathbb{R}$ is $$h(x):=\int_0^x\frac{1}{X(s)}ds \\ .$$ Thus if $X$ is of class $C^k$, possibly not $C^{k+1}$, $\phi $ is in any case in $C^{k+1}(\mathbb{R}\times \mathbb{R},\mathbb{R})\\ .$

Consider for instance the case of $M:=\mathbb{R}$, and any $C^k$ field $X$ on $M$, that we may identify with a function. Say, such that $0 < a \le X(s)\le b$. The field $X$ has a globally defined flow, conjugated to the translation flow $x\mapsto x+t$: $$\phi^t(x):= h^{-1}(h(x)+t)\,,$$ where the conjugation $h:\mathbb{R}\to\mathbb{R}$ is $$h(x):=\int_0^x\frac{1}{X(s)}ds \, .$$ Thus if $X$ is of class $C^k$, possibly not $C^{k+1}$, $\phi $ is in any case in $C^{k+1}(\mathbb{R}\times \mathbb{R},\mathbb{R})\, .$

Consider for instance the case of $M:=\mathbb{R}$, and any $C^k$ field $X$ on $M$, that we may identify with a function. Say, such that $0 < a \le X(s)\le b$. The field $X$ has a globally defined flow conjugated to the flow by translation: $$\phi^t(x):= h^{-1}(h(x)+t)\\ ,$$ where the conjugation is $$h(x):=\int_0^x\frac{1}{X(s)}ds .$$ Thus if $X$ is $C^k(\mathbb{R}, \mathbb{R})$ then $\phi $ is $C^{k+1}(\mathbb{R}\times\mathbb{R}, \mathbb{R})$ even if $X$ was not.

Consider for instance the case of $M:=\mathbb{R}$, and any $C^k$ field $X$ on $M$, that we may identify with a function. Say, such that $0 < a \le X(s)\le b$. The field $X$ has a globally defined flow, conjugated to the translation flow $x\mapsto x+t$: $$\phi^t(x):= h^{-1}(h(x)+t)\\ ,$$ where the conjugation $h:\mathbb{R}\to\mathbb{R}$ is $$h(x):=\int_0^x\frac{1}{X(s)}ds \\ .$$ Thus if $X$ is of class $C^k$, possibly not $C^{k+1}$, $\phi $ is in any case in $C^{k+1}(\mathbb{R}\times \mathbb{R},\mathbb{R})\\ .$