I have a function $P: \mathbb{R}^n \to \mathbb{R}^n$. This function satisfies:
$$ P(\vec{x}) = J\_P(\vec{x}) \cdot \vec{x}$$
where $\vec{x}\in \mathbb{R}^n$, $J\_P$ is the Jacobian of $P$ and "$\cdot$" is
the matrix-vector product. I would rougly describe it as in the title.


I guess that it is a well known property. Do you know what is its name? Can you 
suggest some literature about it, or any known property?


Also, I would like to to know about the symplectic equivalent:
$$ P(\vec{x}) = -\Omega \cdot J\_P(\vec{x}) \cdot \Omega \cdot \vec{x}$$
where $\Omega$ defines the symplectic form.



A function in $\mathbb{R}^n$ is equal to its linearization in each point
As for the symplectic case. Let $J$ be the symplectic matrix $J:=\left[ \matrix{ 0 & I\_n \\ -I\_n & 0 }\right]$. The characteristic lines for the (first order, linear, partial differential, vector) equation 
$$P(x)=-J \,{\rm d}P(x)J x, \qquad x\in\mathbb{R}^{2n}\setminus\{0\},$$
are the solutions of the ODE $\dot \xi=J\xi$, that is circles $\xi(t)=e^{tJ}x\_0$. Along these characteristic lines, the equation is 
$\partial\_t P(\xi(t))= {\rm d}P(\xi(t))J \xi(t)=JP(\xi(t))$, meaning that $P\circ \xi$ satisfies the same ODE, so that we have that $P(e^{tJ}x)=e^{tJ}P(x)$, for all $t\in\mathbb{R}$ and $x\in\mathbb{R}^{2n}\setminus\{0\}$, is a necessary and sufficient condition, for a differentiable map $P:\mathbb{R}^{2n}\setminus\{0\}\to\mathbb{R}^{2n}$, to satisfy your equation. 
We may use complex notation and identify $\mathbb{R}^{2n}$ with $\mathbb{C}^n$, $J$ with the multiplication by $-i$, and the operators $e^{tJ}$ with multiplication by the complex scalars of modulus $1$ (*Warning*: in doing so, we still consider real differentiability, not complex differentiability, i.e. the Fréchet differential ${\rm d}P(x)$ is an $\mathbb{R}$-linear map, not necessarily $\mathbb{C}$-linear, i.e. not assumed to commute with $J$). The condition then reads: $P$ is an [equivariant map](https://en.wikipedia.org/wiki/Equivariant_map) w.r.to the 
group action of $\mathbb{S}^1$:


$$P(\theta x)=\theta P(x)\quad\text{for all } x\in\mathbb{C}^n\setminus\{0\} \text{ and } \theta\in\mathbb{C} \text{ with } |\theta|=1.$$ 


for instance, in dimension $2$ (i.e. $n=1$) these differentiable maps $P:\mathbb{R}^2\setminus\{0\}\to\mathbb{R}^2$ are exactly those that in polar coordinates write as $P(r e^{it})=e^{it}\phi(r)$, where $\phi:\mathbb{R}\_+\to\mathbb{R}$ is a differentiable map. 


**Rmk** If one takes $J$ to be the identity map, and $P$ differentiable for any $x\ne0$, the same computation gives $P(tx)=tP(x)$ for all $t>0$ and $x\ne0$, that is, $P$ is (positively) homogeneous of degree $1$, (equivariant w.r.to the action of homoteties) If $P$ is also assumed to be defined and differentiable at $0$, this implies $P$ linear, as observed in comments. 


