I want to show the following that there is a bijective correspondence between a certain group action $g$ on $\mathbb{R}$ (I believe that that's called a flow - I don't know much about ODE's, so please keep the explanations at an accessible level) and global solutions of a certain type of differential equations (so talking about $g$ is the same as talking about solving certain ODE's), i.e. I want to show the following two theorems:
"1) Let $g:\mathbb{R}^{n}\times\mathbb{R}\rightarrow\mathbb{R}^{n}$ be a group action of the additive group $\mathbb{R}$ on $\mathbb{R}^{n}$ such that $\partial_{2}g$ exists on the whole domain. Then there exists the ODE x'=\partial_{2}g\,\left(x,0\right)\left(x\right),
such that for a solution $l:\mathbb{R}\rightarrow\mathbb{R}^{n}$ with initial value $x_{0}\in\mathbb{R}^{n}$of the above, we have: $l$ is unique and has the form $l\left(t\right):=g\left(x_{0},t\right)$.
(Note, that $\partial_{2}g$ denotes the vector of the partial derivates with respect to the time/the $n+1$'th argument of $\phi$).
2) Conversely if an ODE x'=f\left(x\right) has unique global solutions, then there exists a mapping $g:\mathbb{R}^{n}\times\mathbb{R}\rightarrow\mathbb{R}^{n}$, such that $g$ is a group action ( like described above) and $\partial_{2}g\,\left(x,0\right)=f\left(x\right)$ holds."
My questions are: 1) Have I even stated this proposition correctly ? Is it true what I'm asking ?
2) How can I prove that ? For 1) for example it is easy to show (thanks to the properties of the group action) that if $l$ has the above form, that it is a solution. But taking an arbitrary solution and proving that $l$ has to be of the above form seems to be very difficult.