|\geq 3} d^2_{\beta}(y)\Delta x^{\beta}\right) \left(1-\sum_i\beta_i\right)\\ 
\\
=&\left( \sum_i  \sum_{|\alpha|\geq 1}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha}\right)\left(  \lambda^i_0\right)^2\Delta x_i^2+\\
\\
&+\left(\sum_i \sum_{|\alpha|\geq 1}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha} \right)\left( \lambda^i_0\sum_{|\beta| \geq 3}d^2_{\beta}(y)\beta_i\Delta x^{\beta}\right) \\
\\
&+\left( \sum_i  \sum_{\alpha}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha}\right)\times\\
\\
&\left(  \sum_{|\beta|\geq 3,|\gamma|\geq 3}\beta_i \gamma_i d^2_{\beta}(y)d^2_{\gamma}(y)\Delta x^ {\beta\dot-1}\Delta x^ {\gamma\dot-1}\right).
\end{array}$$

This leads to $$\begin{array}{ll}
&\sum_{|\beta |\geq 3} d^2_{\beta}(y)\Delta x^{\beta}\\ 
\\
=&\frac{1}{\left(1-\sum_i\beta_i\right)}{\Bigg (}\left( \sum_i  \sum_{|\alpha|\geq 1}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha}\right)\left(  \lambda^i_0\right)^2\Delta x_i^2+\\
\\
&+\left(\sum_i \sum_{|\alpha|\geq 1}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha} \right)\left( \lambda^i_0\sum_{|\beta| \geq 3}d^2_{\beta}(y)\beta_i\Delta x^{\beta}\right) \\
\\
&+\left( \sum_i  \sum_{\alpha}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha}\right)\times\\
\\
&\left(  \sum_{|\beta|\geq 3,|\gamma|\geq 3}\beta_i \gamma_i d^2_{\beta}(y)d^2_{\gamma}(y)\Delta x^ {\beta\dot-1}\Delta x^ {\gamma\dot-1}\right){\Bigg )}.
\end{array}$$ Simplifying and renaming multiindices in order to collect for multiindices of order $\beta$ we get $$\begin{array}{ll}
&\sum_{|\beta |\geq 3} d^2_{\beta}(y)\Delta x^{\beta} \\ 
\\
=&\frac{1}{\left(1-\sum_i\beta_i\right)}{\Bigg (} \sum_i  \sum_{|\alpha|\geq 1}\left(  \lambda^i_0\right)^2\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha+2_i}+\\
\\
&+\sum_i \sum_{|\alpha|\geq 1}\sum_{|\gamma| \geq 3}\frac{\lambda_i^{\alpha}}{\alpha!}  \lambda^i_0 d^2_{\gamma}(y)\gamma_i\Delta x^{\alpha +\gamma} \\
\\
&+ \sum_i  \sum_{\alpha}\sum_{|\delta|\geq 3,|\gamma|\geq 3}\frac{\lambda_i^{\alpha}}{\alpha!}  \delta_i \gamma_i d^2_{\delta}(y)d^2_{\gamma}(y)\Delta x^ {\alpha+\gamma+\delta\dot-2_i}{\Bigg )}.
\end{array}$$ The latter equation leads directly to .

Let us draw some consequences out of our theoretical considerations. There is neither an explicit solution nor leads a power series ansatz to a global solution in general. Neither does it help to have local solutions in terms of power series. Such representations are not sufficient for our purposes, since we 