In the paper Dynamics of Extended Bodies in General Relativity. I. Momentum and Angular Momentum, W.G. Dixon proposes definitions for momentum and angular momentum of a certain distribution of matter in GR described by an energy momentum tensor $T$ and a current one-form $J$ on spacetime $(M,g)$.
Essentialy, he defines $W = \operatorname{supp}T\cup \operatorname{supp}J$ as the so-called "world tube" of the matter and supposes that: if $\Sigma \subset M$ is one spacelike hypersurface such that $W\cap \Sigma\neq \emptyset$ then there is one open set $N_\Sigma\subset M$ such that $W\cap \Sigma\subset N_\Sigma$ and $N_\Sigma$ is a normal neighborhood of all its points.
Under these conditions he defines:
$$p^{\kappa}(z,\Sigma)=\int_\Sigma (K^\kappa_\alpha(x,z)\mathfrak{T}^{\alpha\beta}(x)+\Psi^{\kappa}(x,z)\mathfrak{J}^{\beta}(x))n_\beta(x)d^{n-1}x$$
$$S^{\kappa\lambda}(z,\Sigma)=2\int_\Sigma \sigma^{[\lambda}(H^{\kappa]}_\alpha(z,x)\mathfrak{T}^{\alpha\beta}(x)+\Phi^{\kappa]}(x)\mathfrak{J}^\beta(x))n_\beta(x)d^{n-1}x$$
where $\Sigma\subset M$ is a spacelike hypersurface, $n$ is its normal vector, with $z\in \Sigma\cap W \neq \emptyset$. Also $K^{\kappa}_\alpha$, $H^\kappa_\alpha$, $\Psi^\kappa$, $\Phi^\kappa$ are two-point tensors whose definitions don't really seem to matter here.
The issue is that he derives that on maximally symmetric spacetimes (those with constant curvature), given a one-paramter family of hypersurfaces $\Sigma(s)$, $s\in (a,b)$ and a path $\gamma : (a,b)\to M$ with $\gamma(s)\in \Sigma(s)$ the tensor fields over $\gamma$ defined by
$$p^{\kappa}(s)=p^{\kappa}(\gamma(s),\Sigma(s)),\quad S^{\kappa \lambda}(s)=S^{\kappa\lambda}(\gamma(s),\Sigma(s))$$
satisfy the differential equations (with $k$ the constant curvature)
$$\dfrac{D}{ds}p^{\kappa}=k S^{\kappa\lambda}\dot{\gamma}_{\lambda},\quad \dfrac{D}{ds}S^{\kappa\lambda}=2p^{[\kappa}\dot{\gamma}^{\lambda]}.$$
Now the author of the paper says in his words:
The pair of equations (5.7) and (5.9) [those above] can now be integrated along $L$ given the values of $p^\kappa$ and $S^{\kappa\lambda}$ at one point of it. This shows that $p^{\kappa}$ and $S^{\kappa\lambda}$ must be independent of the particular choice of $\Sigma$, depending only on the point $z$ at which they are evaluated. They are thus well defined tensor fields on $M$.
Now I can't understand. He says that those equations implies that the definitions he gave are actually independent of $\Sigma$, i.e., $p^{\kappa}(z,\Sigma)=p^{\kappa}(z,\Sigma')$ and $S^{\kappa\lambda}(z,\Sigma)=S^{\kappa\lambda}(z,\Sigma')$ for distinct $\Sigma,\Sigma'$ with $z\in \Sigma\cap \Sigma'\cap W$.
How to understand what the auhtor says? Why these equations implies independence of $\Sigma$? I thought that it is because if I pick the same curve with two choices of $\Sigma,\Sigma'$, the differential equation is the same, but if I pick another curve connecting two points, why would it be the same, for example?
What really is the point here that solving this equation on the curve implies indepdence of $\Sigma$?
The key point is the uniqueness theorem for ODEs: along any curve $\gamma(s)$ starting at $x_0$ we know that $X(s) = (p(\gamma(s)),S(\gamma(s)))$ satisfies some first-order linear ODE system $X'(s) = L(s) X(s)$, and thus $X$ is determined along the whole curve by its value at $x_0$. Since any point can be connected to $x_0$ by some curve, we can thus determine $p$ and $S$ everywhere by knowing their values at the single point $x_0$.