null() always returns the same number of solutions, namely one. The single solution is the list of vectors that span the null space of the given space. The number of vectors required to span the null space can vary, but each of them is part of the single solution, and it is meaningless to try to force the null space to only have a particular number of vectors.
Note: my computation is that the roots of FF are independent of HH -- that HH acts as a common multiple. The determinant of A is
x^4*HH - x^3*HH - 3*x^2*HH
which is
HH * x^2 * (x^2 - x - 3)
and the roots of that are going to be 0, 0, and 1/2 +/- sqrt(13)/2
What happens after that depends on which of the four roots you happen to get in the 4th position. The documentation for roots() indicates that it uses eig() on a companion matrix, but the order that eig() produces its values in has never been documented and is known to vary between versions; roots() does not impose any further sorting order on the roots.
If one of the 0's happens to end up in the 4th position of the roots() output then the subs() reduces A to [-1 0;0 0] which has the null space [0;1]
If the root 1/2 + sqrt(13)/2 happens to be the one chosen, then the null space will come out as [-(1+sqrt(13))*HH/(-1+sqrt(13); 1] and if the root 1/2 - sqrt(13)/2 happens to be the one chosen, then the null space will come out as [-(1+sqrt(13))*HH/(+1+sqrt(13); 1]
Overall, once you have worked through the math, the only point in going through the loop over HH would be to act as a random number selector over which of these three possibilities would be chosen.
