Making abother system of differential equations with the same choosen solutions

Here I have one problem which should help me to understand how to transform the system of differential equations with the condition to neglect two of four solutions (positive) and to get the appropriate system of equations with reduced order of differential equations but with exactly the same two solutions with two obtained positive chosen solutions of source system.

The source system has the following form

system 1)

g1[x]==a1*X1[x] + a2*X2[x] + a3*Derivative[2][X1][x] - a4*Derivative[2][X2][x] + a5*Derivative[4][X1][x]; second eq

g2[x]==b2*X2[x] + b1*X1[x] - b4*Derivative[2][X2][x] + b3*Derivative[2][X1][x] + b5*Derivative[4][X2][x];

My question is how to transform non-homogenous system into appropriate reduced system with the

system 1) but in this form:

first new eq

AA1*X1[x] + AA2*X2[x] + AA3*Derivative[2][X1][x]=QQ1*g1[x]; second new eq

BB1*X2[x] + BB2*X1[x] + BB3*Derivative[2][X2][x]=QQ2*g2[x];

How to get new non-homogenous system with condition that 2 chosen solutions of source system 1 will be also solutions of new system with reduced order

The problem is how to find constants AA1, AA2, AA3, BB1, BB2, BB3, QQ1 and QQ2 in function of constants a1, a2, b1, b2, a3, b3, a4, b4, a5 and b5

not to solve new system, just to find another appropriate system of equations but in the new form. Two positive solutions in new system vanish (4 in general because of +/-)

what I want. I understand that source system has 8 solutions and new system has 4, but they have nature in the form +/- so I am speaking just about positive solutions. Constants

are known a1, a2, ... Let's suppose that first system has 8 solutions of homogenous part +/- c1*e^(i*v1*t), +/- c2*e^(i*v2*t),+/- c3*e^(i*v3*t), +/- c4*e^(i*v4*t) I want that new

system has solutions +/- c1*e^(i*v1*t), +/- c2*e^(i*v2*t).