in Proposition \[prop\_consistent\] and Lemma \[lem\_n\_zero\] below.

[*Step 4: Application of the implicit function theorem.*]{} The aim is to apply the implicit function theorem on Banach spaces, cf. for example [@d85 Theorem 15.1]. This theorem implies the existence of $\delta>0$ such that there exists a mapping $\mathfrak Z : [0,\delta) \times (-\delta, \delta) \to \mathcal X$ such that for all $(\gamma,\lambda)\in [0,\delta)\times (-\delta,\delta)$ we have $$\mathfrak F(\mathfrak Z(\gamma,\lambda); \gamma, \lambda) = 0,$$ i.e. $\mathfrak Z(\gamma,\lambda)$ is a solution of the reduced EVM-system with parameters $\gamma, \lambda$. This solution $\mathfrak Z(\gamma,\lambda)$ then gives rise to a solution of the EVM-system with the asserted properties, by Proposition \[prop\_equivalent\].

The implicit function theorem can be applied in this way if the operator $\mathfrak F$ is continuous at $(\zeta_0; 0, 0)$, if its Fréchet derivative $\mathfrak L := D\mathfrak F[\zeta_0;0,0]: \mathcal X \to \mathcal X$ at the point $(\zeta_0; 0, 0)\in \mathcal X \times [0,\delta) \times (-\delta,\delta)$ exists and is continuous, and if this Fréchet derivative $\mathfrak L$ is a bijection. These properties are established in Section \[sect\_frechet\_dir\]. Proposition \[prop\_appli\] below contains the details how it is made sure that the boundary conditions for an asymptotically flat solutions are satisfied.

[*Step 5: Characterisation of 