Summary: Non-uniform group divisible designs (GDDs) and non-uniform Kirkman frames are useful in the constructions for other types of designs. In this paper, we consider the existence problems for \(K_{1(3)}\)-GDDs of type \(g^u m^1\) with \(K_{1(3)}=\{k : k \equiv 1 \bmod 3 \}\) and Kirkman frames of type \(h^u m^1\). First, we determine completely the spectrum for uniform \(K_{1(3)}\)-GDDs of type \(g^u\). Then, we consider the entire existence problem for non-uniform \(K_{1(3)}\)-GDDs of type \(g^u m^1\) with \(m > 0\). We show that, for each given \(g\), up to a small number of undetermined cases of \(u\), the necessary conditions on \((u, m)\) for the existence of a \(K_{1(3)}\)-GDD of type \(g^u m^1\) are also sufficient, except possibly when \(u \equiv 2 \bmod 4\) for \(g \equiv 3 \bmod 6\) and when \(u \equiv 6 \bmod 12\) for \(g \equiv 1, 5 \bmod 6\). Finally, a similar result for Kirkman frames of type \(h^u m^1\) is obtained. We show that, for each given \(h\), up to a small number of undetermined cases of \(u\), the necessary conditions on \((u, m)\) for the existence of a Kirkman frame of type \(h^u m^1\) are also sufficient, except possibly when \(u \equiv 2 \bmod 4\) for \(h \equiv 6 \bmod 12\) and when \(u \equiv 6 \bmod 12\) for \(h \equiv 2, 10 \bmod 12\).