Reduce rank of parametric matrix?

Questions
1

Hello,
I need to reduce rank of matrix -6*6 matrix - below from 5 to 2? in the following I attached the Mathematica file.

Thanks

2

Are you certain this is possible? I can find substitutions that bring the rank to 1, 3, or 4, but not to 2.

3

This is a matrix of the homogeneous equation x’=Ax, and I want to find it’s eigenvectors. here, I have defective eigenvalues with defect p and -p. I should find two linearly independent vectors which satisfy these equations:
v0=(A+ lambda I)^2 =0,
v1=(A+ lambda I) v0,
I =denoting a 6*6 identity matrix,
lambda = p or -p,
and Power in the first equation is a rank generalized eigenvector associated with lambda.
Could you please help to find substitutions that bring the rank 1, 3 , 4 of 2 if it is possible?

Thanks