Can we find the parametric equations that sets $A(x) \in \Bbb R^n$ when $A(F(a))=a$ ($F(a) \in \Bbb R$) and $A'= (\nabla F/||\nabla F||^2) \circ A$?

Question

Given

1. a number $n \in \Bbb N^*$,

2. a $n$-tuple $a \in \Bbb R^n$ and

3. a continuous function $F: \space \space \Bbb R^n \longrightarrow \Bbb R$
   $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space x \longmapsto F(x)$,

so "$\nabla F: \Bbb R^n \rightarrow \Bbb R^n$" and "$||\nabla F||^{-2}: \Bbb R^n \rightarrow \Bbb R$" (mapped values can be scalar multiplied),

let $A: \space \space \Bbb R \longrightarrow \Bbb R^n \space \space \space \space \space \space$ such that $(A \circ F)(a)=a$ and $A'= (||\nabla F||^{-2}*\nabla F) \circ A$.
$\space \space \space \space \space \space \space \space \space \space \space \space \space \space x \longmapsto A(x)$

Questions:

1. Is $F \circ A$ identity?
2. Can we systematically find $A$? If only using numerical method, what method?