Hello, I believe I have most of a homogeneous linear system solved. The coefficient matrix is:

That amounts the the following system:

1x +0y + 0w + 0z = 0

0z + 1y + 1w + 0z = 0

Which is:

x = 0

y + w = 0

I let the parameter t = w and got:

x = 0

y = -t

w = t

z = ?

I think z equals another parameter, perhaps s, but I am not certain.