The problem of zero assignment by squaring down is considered for a system of p-inputs, n-outputs and n-states (m > p), where not all outputs are free variables for design. We consider the case where a k-subset of outputs is preserved in the new output set, and the rest are recombined to produce a total new set of p-outputs. New invariants for the problem are introduced which include a new class of fixed zeros and the methodology of the global linearization, developed originally for the output feedback pole assignment problem, is applied to this restricted form of the squaring down problem. It is shown that the problem can be solved generically if (p - k)(m - p) > delta*, where k (k < p) is the number of fixed outputs and * is a system and compensation scheme invariant, which is defined as the restricted Forney degree.