How to find non-trivial of a system of equation?


I have a system of four equations

 eqs4 := {(-5 + 2 c) E^(      4 I c \[Pi]) (E^(4 I c \[Pi]) Subscript[x, 1] - Subscript[x,         4]) == (5 + 2 c) E^(      12 I a c) (Subscript[x, 2] -         E^(4 I c \[Pi]) Subscript[x, 3]), (-5 + 2 c) E^(      6 I a c) (-E^(I b) + E^(6 I a c)) Subscript[x,       3] == (5 + 2 c) (1 - E^(I (b + 6 a c))) Subscript[x,       4], (-5 + 2 c) (-E^(2 I c (3 a + 2 \[Pi])) + E^(        I (b + 8 c \[Pi]))) Subscript[x,       2] == (5 + 2 c) (-E^(-6 I c (a - 2 \[Pi])) + E^(        I (b + 8 c \[Pi]))) Subscript[x, 1],     Subscript[x, 3] + 2 c (Subscript[x, 1] + Subscript[x, 3]) +       Subscript[x, 4] - Subscript[x,       1] - (5 + 2 c) Subscript[x, 2] == -2 c Subscript[x, 4]}; 

and I want to find the non-trivial solution of the system when $ det=0$ . I want to obtain $ \left\{x_1,x_2,x_3,x_4\right\}$ in terms of other parameters.

Trying solve, I only obtain trivial solution zero:

Solve[eqs4, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3],     Subscript[x, 4]}] //   Simplify[#,     Assumptions ->      c \[Element] Reals &&  c > 0  && b \[Element] Reals &&  b > 0 &&       a \[Element] Reals &&  a > 0  && d \[Element] Reals] & 

Then, I try to ignore one of the equation and variables as follows

eqs3 := {(-5 + 2 c) E^(6 I a c) (-E^(I b) + E^(6 I a c)) Subscript[x,       3] == (5 + 2 c) (1 - E^(I (b + 6 a c))) Subscript[x,       4], (-5 + 2 c) (-E^(2 I c (3 a + 2 \[Pi])) + E^(        I (b + 8 c \[Pi]))) Subscript[x,       2] == (5 + 2 c) (-E^(-6 I c (a - 2 \[Pi])) + E^(        I (b + 8 c \[Pi]))) Subscript[x, 1],     Subscript[x, 3] + 2 c (Subscript[x, 1] + Subscript[x, 3]) +       Subscript[x, 4] - Subscript[x,       1] - (5 + 2 c) Subscript[x, 2] == -2 c Subscript[x, 4]}; 

and I get a solution

solution =    Solve[eqs3, {Subscript[x, 2], Subscript[x, 3], Subscript[x, 4]}] //     Simplify[#,       Assumptions ->        c \[Element] Reals &&  c > 0  && b \[Element] Reals &&  b > 0 &&         a \[Element] Reals &&  a > 0  && d \[Element] Reals] &;  

But when I try to verify this solution, one of them is not verified.

eqs4 /. solution //   Simplify[#,     Assumptions ->      c \[Element] Reals &&  c > 0  && b \[Element] Reals &&  b > 0 &&       a \[Element] Reals &&  a > 0  && d \[Element] Reals] & 

Does anybody have a suggestion on how I can find a solution of this system?