= Solution
Resolve the forcing in the eigenbasis:
$$
\begin{pmatrix}1\\a\end{pmatrix}
=(3a-1)u_1+(1-2a)u_2.
$$
In the zero-eigenvalue direction, a constant forcing produces the particular integral $(3a-1)t\,u_1$. In the unit-eigenvalue direction, a constant particular integral is $-(1-2a)u_2$. Thus a particular integral depends on time exactly when
$$
\boxed{a\ne\frac13}.
$$
Adding the complementary solution from part (a), the general solution is
$$
\boxed{
z(t)=\alpha u_1+\beta e^tu_2
+(3a-1)t\,u_1-(1-2a)u_2}.
$$
This is the eigenvector form of the usual <variation of parameters> calculation.
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