How to solve the non-linear over-determined equations symbolically? [closed]

eq1 = 2 *(g3* l1^22 + g4*l22 + g5*l32 -l1* v)*a0 + g1*l12*a02 + 4*a2*g2* k2*l14*R == 0;
eq2 = 2 *(g3*l1^2 + g4* l2^2 + g5* l3^2 - l1*v)*a1 + 2*a0*a1*g1*l1^2 + 2*g2*k^2*l1^4*Q == 0;
eq3 = 2 *(g3* l1^2 + g4* l2^2 + g5*l3^2 - l1*v)*a2 + g1*l1^2*(a1^2 + 2*a0*a2) +
8*a2*g2*k^2*l1^4*Q == 0;
eq4 = 2 a1* a2*g1*l1^2 + 4*a1* g2* k^2*l1^4*P == 0;
eq5 =  a2^2 *g1*l1^2 + 12*a2* g2*k^2*l1^4*P == 0;
aa  = Solve[{eq1, eq2, eq3, eq4, eq5}, { a0, a2, a1}]


I want to solve the overdetermined system symbolically to find a0,a2 and a1. But when I solve it I got empty set. Why?

• An empty set is correct. An overdetermined system does not have a general solution. A solution exists only for special equations. Commented Sep 17, 2023 at 19:38
• What about the a02 from the first equation? Is it unknown or parameter? From eq4 and eq5 follows that $a_1=0$ and $a_2=-\frac{12 \text{g2} k^2 \text{l1}^2 P}{\text{g1}}$. This can be substituted into eq2 to yield $2 \text{g2} k^2 \text{l1}^4 Q=0$. Not sure if it is useful, but you can further analyse in the same way. Commented Sep 17, 2023 at 21:14

1 Answer

Use SolveAlways instead of Solve as follows

eq1 = 2*(g3*l1^22 + g4*l22 + g5*l32 - l1*v)*a0 + g1*l12*a02 +
4*a2*g2*k2*l14*R == 0;
eq2 = 2*(g3*l1^2 + g4*l2^2 + g5*l3^2 - l1*v)*a1 + 2*a0*a1*g1*l1^2 +
2*g2*k^2*l1^4*Q == 0;
eq3 = 2*(g3*l1^2 + g4*l2^2 + g5*l3^2 - l1*v)*a2 +
g1*l1^2*(a1^2 + 2*a0*a2) + 8*a2*g2*k^2*l1^4*Q == 0;
eq4 = 2 a1*a2*g1*l1^2 + 4*a1*g2*k^2*l1^4*P == 0;
eq5 = a2^2*g1*l1^2 + 12*a2*g2*k^2*l1^4*P == 0;
aa = SolveAlways[{eq1, eq2, eq3, eq4, eq5}, {a0, a2, a1}]

(*Out[]= {{a02 -> 0, g5 -> 0, g4 -> 0, g2 -> 0, l1 -> 0}, {a02 -> 0,
k2 -> 0, g5 -> 0, g4 -> 0, l1 -> 0}, {a02 -> 0, l14 -> 0, g5 -> 0,
g4 -> 0, l1 -> 0}, {a02 -> 0, R -> 0, g5 -> 0, g4 -> 0,
l1 -> 0}, {g1 -> 0, g5 -> 0, g4 -> 0, g2 -> 0, l1 -> 0}, {g1 -> 0,
g5 -> 0, g4 -> 0, g2 -> 0, l1 -> 0}, {g1 -> 0, k2 -> 0, g5 -> 0,
g4 -> 0, l1 -> 0}, {g1 -> 0, l14 -> 0, g5 -> 0, g4 -> 0,
l1 -> 0}, {g1 -> 0, R -> 0, g5 -> 0, g4 -> 0, l1 -> 0}, {k2 -> 0,
l12 -> 0, g5 -> 0, g4 -> 0, l1 -> 0}, {l12 -> 0, g5 -> 0, g4 -> 0,
g2 -> 0, l1 -> 0}, {l12 -> 0, l14 -> 0, g5 -> 0, g4 -> 0,
l1 -> 0}, {l12 -> 0, R -> 0, g5 -> 0, g4 -> 0, l1 -> 0}, {a02 -> 0,
k2 -> 0, l32 -> 0, g4 -> 0, l3 -> 0, l1 -> 0}, {a02 -> 0, l14 -> 0,
l32 -> 0, g4 -> 0, l3 -> 0, l1 -> 0}, {a02 -> 0, l32 -> 0, g4 -> 0,
g2 -> 0, l3 -> 0, l1 -> 0}, {a02 -> 0, R -> 0, l32 -> 0, g4 -> 0,
l3 -> 0, l1 -> 0}, {g1 -> 0, g2 -> 0, g4 -> 0, g5 -> 0, l1 -> -1,
v -> -g3}, {g1 -> 0, g2 -> 0, g4 -> 0, g5 -> 0, l1 -> -I,
v -> -I g3}, {g1 -> 0, g2 -> 0, g4 -> 0, g5 -> 0, l1 -> I,
v -> I g3}, {g1 -> 0, g2 -> 0, g4 -> 0, g5 -> 0, l1 -> 1,
v -> g3}, {g1 -> 0, g2 -> 0, g4 -> 0, g5 -> 0, l1 -> -(-1)^(1/10),
v -> -(-1)^(1/10) g3}, {g1 -> 0, g2 -> 0, g4 -> 0, g5 -> 0,
l1 -> (-1)^(1/10), v -> (-1)^(1/10) g3}, {g1 -> 0, g2 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(1/5), v -> -(-1)^(1/5) g3}, {g1 -> 0,
g2 -> 0, g4 -> 0, g5 -> 0, l1 -> (-1)^(1/5),
v -> (-1)^(1/5) g3}, {g1 -> 0, g2 -> 0, g4 -> 0, g5 -> 0,
l1 -> -(-1)^(3/10), v -> -(-1)^(3/10) g3}, {g1 -> 0, g2 -> 0,
g4 -> 0, g5 -> 0, l1 -> (-1)^(3/10), v -> (-1)^(3/10) g3}, {g1 -> 0,
g2 -> 0, g4 -> 0, g5 -> 0, l1 -> -(-1)^(2/5),
v -> -(-1)^(2/5) g3}, {g1 -> 0, g2 -> 0, g4 -> 0, g5 -> 0,
l1 -> (-1)^(2/5), v -> (-1)^(2/5) g3}, {g1 -> 0, g2 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(3/5), v -> -(-1)^(3/5) g3}, {g1 -> 0, g2 -> 0,
g4 -> 0, g5 -> 0, l1 -> (-1)^(3/5), v -> (-1)^(3/5) g3}, {g1 -> 0,
g2 -> 0, g4 -> 0, g5 -> 0, l1 -> -(-1)^(7/10),
v -> -(-1)^(7/10) g3}, {g1 -> 0, g2 -> 0, g4 -> 0, g5 -> 0,
l1 -> (-1)^(7/10), v -> (-1)^(7/10) g3}, {g1 -> 0, g2 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(4/5), v -> -(-1)^(4/5) g3}, {g1 -> 0,
g2 -> 0, g4 -> 0, g5 -> 0, l1 -> (-1)^(4/5),
v -> (-1)^(4/5) g3}, {g1 -> 0, g2 -> 0, g4 -> 0, g5 -> 0,
l1 -> -(-1)^(9/10), v -> -(-1)^(9/10) g3}, {g1 -> 0, g2 -> 0,
g4 -> 0, g5 -> 0, l1 -> (-1)^(9/10), v -> (-1)^(9/10) g3}, {g1 -> 0,
k2 -> 0, l32 -> 0, g4 -> 0, l3 -> 0, l1 -> 0}, {g1 -> 0, l14 -> 0,
l32 -> 0, g4 -> 0, l3 -> 0, l1 -> 0}, {g1 -> 0, l32 -> 0, g4 -> 0,
g2 -> 0, l3 -> 0, l1 -> 0}, {g1 -> 0, l32 -> 0, g4 -> 0, g2 -> 0,
l3 -> 0, l1 -> 0}, {g1 -> 0, R -> 0, l32 -> 0, g4 -> 0, l3 -> 0,
l1 -> 0}, {k2 -> 0, l12 -> 0, l32 -> 0, g4 -> 0, l3 -> 0,
l1 -> 0}, {l12 -> 0, l14 -> 0, l32 -> 0, g4 -> 0, l3 -> 0,
l1 -> 0}, {l12 -> 0, l32 -> 0, g4 -> 0, g2 -> 0, l3 -> 0,
l1 -> 0}, {l12 -> 0, R -> 0, l32 -> 0, g4 -> 0, l3 -> 0,
l1 -> 0}, {g1 -> 0,
l22 -> (g3 l1^2 - g3 l1^22 + g4 l2^2 + g5 l3^2 - g5 l32)/g4,
v -> (g3 l1^2 + g4 l2^2 + g5 l3^2)/l1, g2 -> 0}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, k2 -> 0, l1 -> -1, v -> -g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, k2 -> 0, l1 -> -I, v -> -I g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, k2 -> 0, l1 -> I, v -> I g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, k2 -> 0, l1 -> 1, v -> g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, k2 -> 0, l1 -> -(-1)^(1/10),
v -> -(-1)^(1/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, k2 -> 0,
l1 -> (-1)^(1/10), v -> (-1)^(1/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
k -> 0, k2 -> 0, l1 -> -(-1)^(1/5), v -> -(-1)^(1/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, k2 -> 0, l1 -> (-1)^(1/5),
v -> (-1)^(1/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, k2 -> 0,
l1 -> -(-1)^(3/10), v -> -(-1)^(3/10) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, k2 -> 0, l1 -> (-1)^(3/10),
v -> (-1)^(3/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, k2 -> 0,
l1 -> -(-1)^(2/5), v -> -(-1)^(2/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
k -> 0, k2 -> 0, l1 -> (-1)^(2/5), v -> (-1)^(2/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, k2 -> 0, l1 -> -(-1)^(3/5),
v -> -(-1)^(3/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, k2 -> 0,
l1 -> (-1)^(3/5), v -> (-1)^(3/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
k -> 0, k2 -> 0, l1 -> -(-1)^(7/10),
v -> -(-1)^(7/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, k2 -> 0,
l1 -> (-1)^(7/10), v -> (-1)^(7/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
k -> 0, k2 -> 0, l1 -> -(-1)^(4/5), v -> -(-1)^(4/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, k2 -> 0, l1 -> (-1)^(4/5),
v -> (-1)^(4/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, k2 -> 0,
l1 -> -(-1)^(9/10), v -> -(-1)^(9/10) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, k2 -> 0, l1 -> (-1)^(9/10),
v -> (-1)^(9/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, l1 -> -1,
l14 -> 0, v -> -g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, l1 -> -1,
R -> 0, v -> -g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, l1 -> -I,
l14 -> 0, v -> -I g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, l1 -> -I,
R -> 0, v -> -I g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, l1 -> I,
l14 -> 0, v -> I g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, l1 -> I,
R -> 0, v -> I g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, l1 -> 1,
l14 -> 0, v -> g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0, l1 -> 1,
R -> 0, v -> g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> -(-1)^(1/10), l14 -> 0, v -> -(-1)^(1/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> -(-1)^(1/10), R -> 0,
v -> -(-1)^(1/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> (-1)^(1/10), l14 -> 0, v -> (-1)^(1/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> (-1)^(1/10), R -> 0,
v -> (-1)^(1/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> -(-1)^(1/5), l14 -> 0, v -> -(-1)^(1/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> -(-1)^(1/5), R -> 0,
v -> -(-1)^(1/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> (-1)^(1/5), l14 -> 0, v -> (-1)^(1/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, l1 -> (-1)^(1/5), R -> 0,
v -> (-1)^(1/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> -(-1)^(3/10), l14 -> 0, v -> -(-1)^(3/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> -(-1)^(3/10), R -> 0,
v -> -(-1)^(3/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> (-1)^(3/10), l14 -> 0, v -> (-1)^(3/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> (-1)^(3/10), R -> 0,
v -> (-1)^(3/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> -(-1)^(2/5), l14 -> 0, v -> -(-1)^(2/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> -(-1)^(2/5), R -> 0,
v -> -(-1)^(2/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> (-1)^(2/5), l14 -> 0, v -> (-1)^(2/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, l1 -> (-1)^(2/5), R -> 0,
v -> (-1)^(2/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> -(-1)^(3/5), l14 -> 0, v -> -(-1)^(3/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> -(-1)^(3/5), R -> 0,
v -> -(-1)^(3/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> (-1)^(3/5), l14 -> 0, v -> (-1)^(3/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, l1 -> (-1)^(3/5), R -> 0,
v -> (-1)^(3/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> -(-1)^(7/10), l14 -> 0, v -> -(-1)^(7/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> -(-1)^(7/10), R -> 0,
v -> -(-1)^(7/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> (-1)^(7/10), l14 -> 0, v -> (-1)^(7/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> (-1)^(7/10), R -> 0,
v -> (-1)^(7/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> -(-1)^(4/5), l14 -> 0, v -> -(-1)^(4/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> -(-1)^(4/5), R -> 0,
v -> -(-1)^(4/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> (-1)^(4/5), l14 -> 0, v -> (-1)^(4/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, k -> 0, l1 -> (-1)^(4/5), R -> 0,
v -> (-1)^(4/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> -(-1)^(9/10), l14 -> 0, v -> -(-1)^(9/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> -(-1)^(9/10), R -> 0,
v -> -(-1)^(9/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k -> 0,
l1 -> (-1)^(9/10), l14 -> 0, v -> (-1)^(9/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k -> 0, l1 -> (-1)^(9/10), R -> 0,
v -> (-1)^(9/10) g3}, {a02 -> 0, k2 -> 0, l22 -> -((g5 l32)/g4),
l2 -> 0, l3 -> 0, l1 -> 0}, {a02 -> 0, l14 -> 0,
l22 -> -((g5 l32)/g4), l2 -> 0, l3 -> 0, l1 -> 0}, {a02 -> 0,
l22 -> -((g5 l32)/g4), g2 -> 0, l2 -> 0, l3 -> 0,
l1 -> 0}, {a02 -> 0, R -> 0, l22 -> -((g5 l32)/g4), l2 -> 0,
l3 -> 0, l1 -> 0}, {g1 -> 0, k2 -> 0, l22 -> -((g5 l32)/g4),
l2 -> 0, l3 -> 0, l1 -> 0}, {g1 -> 0, l14 -> 0,
l22 -> -((g5 l32)/g4), l2 -> 0, l3 -> 0, l1 -> 0}, {g1 -> 0,
l22 -> -((g5 l32)/g4), g2 -> 0, l2 -> 0, l3 -> 0,
l1 -> 0}, {g1 -> 0, l22 -> -((g5 l32)/g4), g2 -> 0, l2 -> 0,
l3 -> 0, l1 -> 0}, {g1 -> 0, R -> 0, l22 -> -((g5 l32)/g4), l2 -> 0,
l3 -> 0, l1 -> 0}, {k2 -> 0, l12 -> 0, l22 -> -((g5 l32)/g4),
l2 -> 0, l3 -> 0, l1 -> 0}, {l12 -> 0, l14 -> 0,
l22 -> -((g5 l32)/g4), l2 -> 0, l3 -> 0, l1 -> 0}, {l12 -> 0,
l22 -> -((g5 l32)/g4), g2 -> 0, l2 -> 0, l3 -> 0,
l1 -> 0}, {l12 -> 0, R -> 0, l22 -> -((g5 l32)/g4), l2 -> 0,
l3 -> 0, l1 -> 0}, {g1 -> 0, k2 -> 0,
l22 -> (g3 l1^2 - g3 l1^22 + g4 l2^2 + g5 l3^2 - g5 l32)/g4,
v -> (g3 l1^2 + g4 l2^2 + g5 l3^2)/l1, k -> 0}, {g1 -> 0, l14 -> 0,
l22 -> (g3 l1^2 - g3 l1^22 + g4 l2^2 + g5 l3^2 - g5 l32)/g4,
v -> (g3 l1^2 + g4 l2^2 + g5 l3^2)/l1, k -> 0}, {g1 -> 0, R -> 0,
l22 -> (g3 l1^2 - g3 l1^22 + g4 l2^2 + g5 l3^2 - g5 l32)/g4,
v -> (g3 l1^2 + g4 l2^2 + g5 l3^2)/l1, k -> 0}, {a02 -> 0, k2 -> 0,
l22 -> (l2^2 l32)/l3^2, g5 -> -((g4 l2^2)/l3^2),
l1 -> 0}, {a02 -> 0, l14 -> 0, l22 -> (l2^2 l32)/l3^2,
g5 -> -((g4 l2^2)/l3^2), l1 -> 0}, {a02 -> 0,
l22 -> (l2^2 l32)/l3^2, g5 -> -((g4 l2^2)/l3^2), g2 -> 0,
l1 -> 0}, {a02 -> 0, R -> 0, l22 -> (l2^2 l32)/l3^2,
g5 -> -((g4 l2^2)/l3^2), l1 -> 0}, {g1 -> 0, k2 -> 0,
l22 -> (l2^2 l32)/l3^2, g5 -> -((g4 l2^2)/l3^2), l1 -> 0}, {g1 -> 0,
l14 -> 0, l22 -> (l2^2 l32)/l3^2, g5 -> -((g4 l2^2)/l3^2),
l1 -> 0}, {g1 -> 0, l22 -> (l2^2 l32)/l3^2, g5 -> -((g4 l2^2)/l3^2),
g2 -> 0, l1 -> 0}, {g1 -> 0, l22 -> (l2^2 l32)/l3^2,
g5 -> -((g4 l2^2)/l3^2), g2 -> 0, l1 -> 0}, {g1 -> 0, R -> 0,
l22 -> (l2^2 l32)/l3^2, g5 -> -((g4 l2^2)/l3^2), l1 -> 0}, {k2 -> 0,
l12 -> 0, l22 -> (l2^2 l32)/l3^2, g5 -> -((g4 l2^2)/l3^2),
l1 -> 0}, {l12 -> 0, l14 -> 0, l22 -> (l2^2 l32)/l3^2,
g5 -> -((g4 l2^2)/l3^2), l1 -> 0}, {l12 -> 0,
l22 -> (l2^2 l32)/l3^2, g5 -> -((g4 l2^2)/l3^2), g2 -> 0,
l1 -> 0}, {l12 -> 0, R -> 0, l22 -> (l2^2 l32)/l3^2,
g5 -> -((g4 l2^2)/l3^2), l1 -> 0}, {g1 -> 0,
l32 -> (g3 l1^2 - g3 l1^22 + g5 l3^2)/g5,
v -> (g3 l1^2 + g5 l3^2)/l1, g4 -> 0, g2 -> 0}, {g1 -> 0, g4 -> 0,
g5 -> 0, k2 -> 0, l1 -> -1, P -> 0, Q -> 0, v -> -g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k2 -> 0, l1 -> -I, P -> 0, Q -> 0,
v -> -I g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k2 -> 0, l1 -> I, P -> 0,
Q -> 0, v -> I g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k2 -> 0, l1 -> 1,
P -> 0, Q -> 0, v -> g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k2 -> 0,
l1 -> -(-1)^(1/10), P -> 0, Q -> 0, v -> -(-1)^(1/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k2 -> 0, l1 -> (-1)^(1/10), P -> 0, Q -> 0,
v -> (-1)^(1/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k2 -> 0,
l1 -> -(-1)^(1/5), P -> 0, Q -> 0, v -> -(-1)^(1/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k2 -> 0, l1 -> (-1)^(1/5), P -> 0, Q -> 0,
v -> (-1)^(1/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k2 -> 0,
l1 -> -(-1)^(3/10), P -> 0, Q -> 0, v -> -(-1)^(3/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k2 -> 0, l1 -> (-1)^(3/10), P -> 0, Q -> 0,
v -> (-1)^(3/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k2 -> 0,
l1 -> -(-1)^(2/5), P -> 0, Q -> 0, v -> -(-1)^(2/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k2 -> 0, l1 -> (-1)^(2/5), P -> 0, Q -> 0,
v -> (-1)^(2/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k2 -> 0,
l1 -> -(-1)^(3/5), P -> 0, Q -> 0, v -> -(-1)^(3/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k2 -> 0, l1 -> (-1)^(3/5), P -> 0, Q -> 0,
v -> (-1)^(3/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k2 -> 0,
l1 -> -(-1)^(7/10), P -> 0, Q -> 0, v -> -(-1)^(7/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k2 -> 0, l1 -> (-1)^(7/10), P -> 0, Q -> 0,
v -> (-1)^(7/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k2 -> 0,
l1 -> -(-1)^(4/5), P -> 0, Q -> 0, v -> -(-1)^(4/5) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k2 -> 0, l1 -> (-1)^(4/5), P -> 0, Q -> 0,
v -> (-1)^(4/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, k2 -> 0,
l1 -> -(-1)^(9/10), P -> 0, Q -> 0, v -> -(-1)^(9/10) g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, k2 -> 0, l1 -> (-1)^(9/10), P -> 0, Q -> 0,
v -> (-1)^(9/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> -1,
l14 -> 0, P -> 0, Q -> 0, v -> -g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
l1 -> -1, P -> 0, Q -> 0, R -> 0, v -> -g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> -I, l14 -> 0, P -> 0, Q -> 0, v -> -I g3}, {g1 -> 0,
g4 -> 0, g5 -> 0, l1 -> -I, P -> 0, Q -> 0, R -> 0,
v -> -I g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> I, l14 -> 0, P -> 0,
Q -> 0, v -> I g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> I, P -> 0,
Q -> 0, R -> 0, v -> I g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> 1,
l14 -> 0, P -> 0, Q -> 0, v -> g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
l1 -> 1, P -> 0, Q -> 0, R -> 0, v -> g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(1/10), l14 -> 0, P -> 0, Q -> 0,
v -> -(-1)^(1/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
l1 -> -(-1)^(1/10), P -> 0, Q -> 0, R -> 0,
v -> -(-1)^(1/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
l1 -> (-1)^(1/10), l14 -> 0, P -> 0, Q -> 0,
v -> (-1)^(1/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> (-1)^(1/10),
P -> 0, Q -> 0, R -> 0, v -> (-1)^(1/10) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(1/5), l14 -> 0, P -> 0, Q -> 0,
v -> -(-1)^(1/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> -(-1)^(1/5),
P -> 0, Q -> 0, R -> 0, v -> -(-1)^(1/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> (-1)^(1/5), l14 -> 0, P -> 0, Q -> 0,
v -> (-1)^(1/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> (-1)^(1/5),
P -> 0, Q -> 0, R -> 0, v -> (-1)^(1/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(3/10), l14 -> 0, P -> 0, Q -> 0,
v -> -(-1)^(3/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
l1 -> -(-1)^(3/10), P -> 0, Q -> 0, R -> 0,
v -> -(-1)^(3/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
l1 -> (-1)^(3/10), l14 -> 0, P -> 0, Q -> 0,
v -> (-1)^(3/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> (-1)^(3/10),
P -> 0, Q -> 0, R -> 0, v -> (-1)^(3/10) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(2/5), l14 -> 0, P -> 0, Q -> 0,
v -> -(-1)^(2/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> -(-1)^(2/5),
P -> 0, Q -> 0, R -> 0, v -> -(-1)^(2/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> (-1)^(2/5), l14 -> 0, P -> 0, Q -> 0,
v -> (-1)^(2/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> (-1)^(2/5),
P -> 0, Q -> 0, R -> 0, v -> (-1)^(2/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(3/5), l14 -> 0, P -> 0, Q -> 0,
v -> -(-1)^(3/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> -(-1)^(3/5),
P -> 0, Q -> 0, R -> 0, v -> -(-1)^(3/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> (-1)^(3/5), l14 -> 0, P -> 0, Q -> 0,
v -> (-1)^(3/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> (-1)^(3/5),
P -> 0, Q -> 0, R -> 0, v -> (-1)^(3/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(7/10), l14 -> 0, P -> 0, Q -> 0,
v -> -(-1)^(7/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
l1 -> -(-1)^(7/10), P -> 0, Q -> 0, R -> 0,
v -> -(-1)^(7/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
l1 -> (-1)^(7/10), l14 -> 0, P -> 0, Q -> 0,
v -> (-1)^(7/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> (-1)^(7/10),
P -> 0, Q -> 0, R -> 0, v -> (-1)^(7/10) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(4/5), l14 -> 0, P -> 0, Q -> 0,
v -> -(-1)^(4/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> -(-1)^(4/5),
P -> 0, Q -> 0, R -> 0, v -> -(-1)^(4/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> (-1)^(4/5), l14 -> 0, P -> 0, Q -> 0,
v -> (-1)^(4/5) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> (-1)^(4/5),
P -> 0, Q -> 0, R -> 0, v -> (-1)^(4/5) g3}, {g1 -> 0, g4 -> 0,
g5 -> 0, l1 -> -(-1)^(9/10), l14 -> 0, P -> 0, Q -> 0,
v -> -(-1)^(9/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
l1 -> -(-1)^(9/10), P -> 0, Q -> 0, R -> 0,
v -> -(-1)^(9/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0,
l1 -> (-1)^(9/10), l14 -> 0, P -> 0, Q -> 0,
v -> (-1)^(9/10) g3}, {g1 -> 0, g4 -> 0, g5 -> 0, l1 -> (-1)^(9/10),
P -> 0, Q -> 0, R -> 0, v -> (-1)^(9/10) g3}, {g1 -> 0, k2 -> 0,
P -> 0, l22 -> (g3 l1^2 - g3 l1^22 + g4 l2^2 + g5 l3^2 - g5 l32)/g4,
v -> (g3 l1^2 + g4 l2^2 + g5 l3^2)/l1, Q -> 0}, {g1 -> 0, l14 -> 0,
P -> 0,
l22 -> (g3 l1^2 - g3 l1^22 + g4 l2^2 + g5 l3^2 - g5 l32)/g4,
v -> (g3 l1^2 + g4 l2^2 + g5 l3^2)/l1, Q -> 0}, {g1 -> 0, P -> 0,
R -> 0, l22 -> (g3 l1^2 - g3 l1^22 + g4 l2^2 + g5 l3^2 - g5 l32)/g4,
v -> (g3 l1^2 + g4 l2^2 + g5 l3^2)/l1, Q -> 0}, {g1 -> 0, k2 -> 0,
l32 -> (g3 l1^2 - g3 l1^22 + g5 l3^2)/g5,
v -> (g3 l1^2 + g5 l3^2)/l1, g4 -> 0, k -> 0}, {g1 -> 0, l14 -> 0,
l32 -> (g3 l1^2 - g3 l1^22 + g5 l3^2)/g5,
v -> (g3 l1^2 + g5 l3^2)/l1, g4 -> 0, k -> 0}, {g1 -> 0, R -> 0,
l32 -> (g3 l1^2 - g3 l1^22 + g5 l3^2)/g5,
v -> (g3 l1^2 + g5 l3^2)/l1, g4 -> 0, k -> 0}, {g1 -> 0, k2 -> 0,
P -> 0, l32 -> (g3 l1^2 - g3 l1^22 + g5 l3^2)/g5,
v -> (g3 l1^2 + g5 l3^2)/l1, g4 -> 0, Q -> 0}, {g1 -> 0, l14 -> 0,
P -> 0, l32 -> (g3 l1^2 - g3 l1^22 + g5 l3^2)/g5,
v -> (g3 l1^2 + g5 l3^2)/l1, g4 -> 0, Q -> 0}, {g1 -> 0, P -> 0,
R -> 0, l32 -> (g3 l1^2 - g3 l1^22 + g5 l3^2)/g5,
v -> (g3 l1^2 + g5 l3^2)/l1, g4 -> 0, Q -> 0}, {g1 -> 0, v -> 0,
g5 -> 0, g4 -> 0, g2 -> 0, g3 -> 0}, {g1 -> 0, k2 -> 0, v -> 0,
g5 -> 0, g4 -> 0, k -> 0, g3 -> 0}, {g1 -> 0, l14 -> 0, v -> 0,
g5 -> 0, g4 -> 0, k -> 0, g3 -> 0}, {g1 -> 0, R -> 0, v -> 0,
g5 -> 0, g4 -> 0, k -> 0, g3 -> 0}, {g1 -> 0, k2 -> 0, P -> 0,
v -> 0, g5 -> 0, g4 -> 0, Q -> 0, g3 -> 0}, {g1 -> 0, l14 -> 0,
P -> 0, v -> 0, g5 -> 0, g4 -> 0, Q -> 0, g3 -> 0}, {g1 -> 0,
P -> 0, R -> 0, v -> 0, g5 -> 0, g4 -> 0, Q -> 0, g3 -> 0}}*)