Find all straight lines lying in S.

One set is the intersection of the planes y=k;z=kx for any real k.

Another is x=k;z=ky for all real k.

Expressed parametrically:

Set one: y=k;x=t;z=kt
Set two: y=t;x=k;z=kt

That there are no others is shown by the fact that for the line to be straight while both x and y vary, y must be in the form ax+b.  Then the equation for z becomes z=x(ax+b) = ax˛+bx, but if a is not zero, this is not linear, so a must be zero (so long as x is not a constant) and y a constant.  Interchange x and y in the above for the other way of looking at why not both x and y can vary.