> > If you are still sure that you have a continuum of solutions
>
>                                                                     thus
> creating the continuum.

OK, reconsidering this, I agree that there probably is a continuum.
In my approach this results from the vector equation being an equation
between _unit_ vectors. Hence the equation essentially only removes two
degrees of freedom.
(Of course, all this assumes that everything is regular: The single
equation x*x+y*y=0 fixes both x and y, so thinking in terms of degrees
of freedom always is just heuristic.)

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