Help in solving nonlinear system of equations

Question

I have a system of six non-linear equations in six unknowns, which I do not know how to solve.

$$\left( 1 + 4 \left( x_2^2 + x_3^2 \right) \right)x_4 - 4x_1x_2x_5 - 4x_1x_3x_6 = 0,$$ $$-4x_1x_2x_4 + \left( 1 + 4 \left( x_1^2 + x_3^2 \right) \right)x_5 - 4x_2x_3x_6 = 0,$$ $$-4x_1x_3x_4 - 4x_2x_3x_5 + \left( 1 + 4 \left( x_1^2 + x_2^2 \right) \right)x_6 = 0,$$ $$\left( 1 - 4 \left( x_5^2 + x_6^2 \right) \right)x_1 + 4x_4x_5x_2 + 4x_4x_6x_3 = 0,$$ $$4x_4x_5x_1 + \left( 1 - 4 \left( x_4^2 + x_6^2 \right) \right)x_2 + 4x_5x_6x_3 = 0,$$ $$4x_4x_6x_1 + 4x_5x_6x_2 + \left( 1 - 4 \left( x_4^2 + x_5^2 \right) \right)x_3 = 0.$$

As a start, I have added the first three equations and the last three equations to try and get a perfect square. In the process, I have:

$$\left( 1 + 2 \left( x_1 + x_2 + x_3 \right)^2 \right) \left( x_4 + x_5 + x_6 \right) - 2 \left( 2x_2x_3 + x_1^2 \right)x_4 - 2 \left( 2x_1x_3 + x_2^2 \right)x_5 - 2 \left( 2x_1x_2 + x_3^2 \right)x_6 = 0,$$ $$\left( 1 - 2 \left( x_4 + x_5 + x_6 \right)^2 \right) \left( x_1 + x_2 + x_3 \right) + 2 \left( 2x_5x_6 + x_4^2 \right)x_1 + 2 \left( 2x_4x_6 + x_5^2 \right)x_2 + 2 \left( 2x_4x_5 + x_6^2 \right)x_3 = 0.$$

From here, I have no idea how to proceed! Is there any general method to solve a system of nonlinear equations? Any help will be appreciated!

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PS: The system is only a part of a system of $90$ equations in $36$ unknowns which I obtained when trying to find a Lie algebra isomorphism between $\mathfrak{so} \left( 3; 1 \right)_{\mathbb{C}}$ and $\mathfrak{sl} \left( 2, \mathbb{C} \right) \oplus \mathfrak{sl} \left( 2, \mathbb{C} \right)$. Most of the equations in the system are of this form, and if I can solve this system, I suppose the other $30$ variables can be known easily.

Answer 1

Using Maxima

load(grobner);
vars : [x1,x2,x3,x4,x5,x6];

eqns : [(1+4*(x2^2+x3^2))*x4−4*x1*x2*x5−4*x1*x3*x6,
−4*x1*x2*x4+(1+4*(x1^2+x3^2))*x5 −4*x2*x3*x6,
−4*x1*x3*x4 −4*x2*x3*x5+(1+4*(x1^2+x2^2))*x6,
(1−4*(x5^2+x6^2))*x1 + 4*x4*x5*x2 +4*x4*x6*x3,
4*x4*x5*x1+(1−4*(x4^2+x6^2))*x2 + 4*x5*x6*x3,
4*x4*x6*x1+4*x5*x6*x2+(1−4*(x4^2+x5^2))*x3];

eqns : expand(eqns);

gb : poly_reduced_grobner(eqns,vars);

$$\begin{pmatrix} -4\,x_{1} \,x_{3}\,x_{6}-4\,x_{1}\,x_{2}\,x_{5}+4\,x_{3}^2\,x_{4}+4\,x_{2}^2\, x_{4}+x_{4}\cr -4\,x_{1}\,x_{6}^2+4\,x_{3}\,x_{4}\,x_{6}-4\,x_{1}\, x_{5}^2+4\,x_{2}\,x_{4}\,x_{5}+x_{1}\cr x_{3}\,x_{6}+x_{2}\,x_{5}+ x_{1}\,x_{4}\cr 4\,x_{1}\,x_{2}\,x_{6}^2-4\,x_{1}\,x_{3}\,x_{5}\, x_{6}-4\,x_{2}\,x_{3}\,x_{4}\,x_{6}+4\,x_{3}^2\,x_{4}\,x_{5}+x_{4}\, x_{5}-x_{1}\,x_{2}\cr -4\,x_{2}^2\,x_{6}^2+8\,x_{2}\,x_{3}\,x_{5}\, x_{6}-4\,x_{3}^2\,x_{5}^2+x_{4}^2+x_{3}^2+x_{2}^2\cr 4\,x_{4}\,x_{6} ^2+4\,x_{4}\,x_{5}^2+4\,x_{4}^3-x_{4}\cr -4\,x_{6}^3-4\,x_{5}^2\, x_{6}-4\,x_{4}^2\,x_{6}+x_{6}\cr -4\,x_{5}\,x_{6}^2-4\,x_{5}^3-4\, x_{4}^2\,x_{5}+x_{5}\cr -4\,x_{3}\,x_{6}^2-4\,x_{3}\,x_{5}^2-4\, x_{3}\,x_{4}^2+x_{3}\cr -4\,x_{2}\,x_{6}^2-4\,x_{2}\,x_{5}^2-4\, x_{2}\,x_{4}^2+x_{2}\cr -x_{6}^2-x_{5}^2-x_{4}^2-x_{3}^2-x_{2}^2- x_{1}^2\cr \end{pmatrix}$$

All these equations are equal to $0$.

The last equation gives $x_{6}^2+x_{5}^2+x_{4}^2+x_{3}^2+x_{2}^2+ x_{1}^2 = 0$

Looking at the equations they seem to be very symmetric.

It implies that some more properties/equations are needed to differentiate the variables.