Recently, I have studied some ODE theory and I am curious about the plane curves in $\mathbb R^n$. Here, a plane curve means a curve that stays on a parametric plane of $\mathbb R^n$, e.g, $P := \{ x \in \mathbb R^n: \sum_{i=1}^n p_i x_i = k \}$, where $k \in \mathbb R$ and not all $p_i, i = 1,...,n$ are 0.
Let $f: \mathbb R^n \rightarrow P$, I would like to study the equation
$$ \dot x_t = f(x_t), \quad t >0, \quad x_0 \in P, \quad (*) $$
such that the solutions $(x_t)_{t\geq 0}$ stay on $P$. A curve $(x_t)_{t\geq 0}$ is a solution to $(*)$ iff $(x_t)_{t\geq 0} \subset P$ and satisfies $(*)$ in the sense that, for $t_0 >0$,
$$ f(x_{t_0}) = \dot x_t|_{t=t_0} := \lim_{t \rightarrow t_0} \frac{x_t - x_{t_0}}{t - t_0} = \left( \lim_{t \rightarrow t_0} \frac{x^1_t - x^1_{t_0}}{t - t_0},..., \lim_{t \rightarrow t_0} \frac{x^n_t - x^n_{t_0}}{t - t_0} \right),$$ where all the limits in the bracket above must exist.
It can be seen that if $f = (f^1,...,f^n)$ then necessarily,
$$ \sum_{i=1}^n p_i f^i(x_t) = 0, \quad \forall t >0,$$
so we can assume $f: \mathbb R^n \rightarrow P^0 := \{ x \in \mathbb R^n: \sum_{i=1}^n p_i x_i = 0\}$ beforehand.
Has there been any studies on the existence and uniqueness of $(*)$ in the literature before? If yes, can you help me find some reference?
There are two ways I see to interpret your question, and both are interesting and well-studied.
The first is investigating systems $\dot{x} = f(x)$ whose solutions stay on a plane. Such systems will satisfy the conditions you have stated. In this case, the plane is called an invariant set of the system $\dot{x} = f(x)$. These are extremely important in the analysis of dynamical systems, as often systems can be reduced to simpler representations within their invariant sets, or combinations of invariant sets and fixed points can give detailed information about the existence of stable manifolds, periodic orbits, etc.
The second is taking the system $\dot{x} = f(x)$ and explicitly constraining it to lie on a specified plane parameterized in this case by $p\cdot x - k = 0$. This results in a system of equations of the form $$ \begin{aligned} \dot{x} &= f(x) \\ 0 &= p\cdot x - k. \end{aligned} $$ This is known as a differential-algebraic equation (DAE) and there is a large amount of literature on their properties and numerical simulation. Obviously, not every initial condition is even compatible with this system. Furthermore, even for compatible initial conditions, the existence and uniqueness theory is more complicated than for ODEs, especially for nonlinear constraints.