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Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let $\mathrm{m}od\left(f\right)=L_1\oplus L_2$ be its frequency module. We say that an almost periodic solution $x\left(t\right)$ of the system $$\dot{x}=f(t,x),t\in ℝ,x\in D\subset {ℝ}^n$$ is irregular with respect to $L_2$ (or partially irregular) if $(\mathrm{m}od\left(x\right)+L_1)\cap L_2=\left\{0\right\}$. Suppose that $f(t,x)=A\left(t\right)x+X(t,x),$ where $A\left(t\right)$ is an almost periodic $(n\times n)$-matrix and $\mathrm{m}od\left(A\right)\cap \mathrm{m}od\left(X\right)=\left\{0\right\}.$ We consider the existence problem for almost periodic irregular with respect to $\mathrm{m}od\left(A\right)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions...