Let \begin{align} \dot x = f(x) \tag{1} \end{align} be an ODE where $f:\mathbb{R}^n \to \mathbb{R}^n$ is smooth. Let us also define the map $M:\mathbb{R}^n\to\mathbb{R}^{n-1}$ that maps $x\mapsto y$ defined as \begin{align} M(x)=\{y \in \mathbb{R}^{n-1} : y_i = x_i - x_n, \forall i \in \{1,...,n-1\} \}. \end{align} Suppose we perform a change of coordinates from $x$ to $y$. Now, what is the relationship between the original ODE (1) and the new ODE \begin{align} \dot y = g(y) \tag{2} \end{align} which is in a lower dimension? Any idea/response is greatly appreciated!
My work: I tried to prove that the two systems (1) and (2) are topologically equivalent. The map $M$ is smooth. However, I am not sure if I can define an inverse for $M$. For instance, we can define \begin{align} M^{-1}(y)=\{x \in \mathbb{R}^n : x_i = y_i, \forall i \in \{1,...,n-1\}, x_n = 0 \}. \end{align} Can we say that this "defined" inverse function is smooth? If yes, can we conclude that the restriction of $M$ to $\mathbb{R}^{n-1}\times\{0\}$ is a homeomorphism?