An underdetermined system of equations for the determination
of the Lyapunov-Schmidt reduction is presented.
This system can easily be extended to an augmented system
for the the computation of nontrivial singularities
by adding suitable equations, which
characterize the singularity.
Necessary and sufficient conditions for
these additional equations are given,
so that the corresponding Gauss-Newton method
converges locally and quadratically.
For several types of singularities possible equations
are derived from invariance conditions.
Numerical examples illustrating the results are included.
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