The Gauss-Codazzi equations govern the geometry of surfaces in R${}^n$.
In 1897, Hazzi\-dakis found a reduction to a codimension three P6 equation in the case $n=3$.
Our motivation is to find a reduction to the full (codimension zero) P6.
Since the Gauss-Codazzi equations are underdetermined (three equations in four unknowns), we first restrict them to a determined system and compute its Lie point symmetries.
This allows us to find three more reductions to P6, with respective codimensions three, two, two.

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{\bf References:}
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\item[1.]
%\bibitem{BE2000}
A.I.~Bobenko and U.~Eitner,
Painlev\'e equations in differential geometry of surfaces,
%120 pages,
Lecture Notes in Math.~{\bf 1753} (2000). % (Springer, Berlin, 2000).
%http://www-sfb288.math.tu-berlin.de

\item[2.]
%\bibitem{CGS1995}
J.L.~Cie\'sli\'nski, P.~Goldstein and A.~Sym,
%Isothermic surfaces in E${}^3$ as soliton surfaces,
Phys.~Lett.~A {\bf 205} (1995) 37--43.

\item[3.]
%\bibitem{CMBook}
R.~Conte and M.~Musette,
\textit{The Painlev\'e handbook} (Springer, Berlin, 2008).
Russian translation (RCD, Moscow, 2011).
% http://www.springer.com/physics/book/978-1-4020-8490-4

\item[4.]
J.N.~Hazzidakis,
%Biegung mit Erhaltung der Hauptkr\"ummungsradien,
Journal f\"ur die reine und angewandte Mathematik,
{\bf 117} (1897) 42?56.


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