Let f(t, x) be a vector valued function almost periodic in t uniformly for x, and let mod(f) = L1 ⊕ L2 be its frequency module. We say that an almost periodic solution x(t) of the system x˙ = f(t, x), t ∈ , x ∈ D ⊂ Rn is irregular with respect to L2 (or partially irregular) if (mod(x) + L1) ∩ L2 = {0}. Suppose that f(t, x) = A(t)x + X(t, x), where A(t) is an almost periodic (n × n)-matrix and mod(A) ∩ mod(X) = {0}. We consider the existence problem for almost periodic irregular with respect to mod(A) solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions for existence of such solutions are obtained.