  The one-dimensional Kondo lattice model (1D KLM) is usually defined by the
Kondo exchange $J$ between conduction electrons and spins of the array, and the
hopping strength t for the moving electrons. Here, we also include a direct
exchange $K$ term between spins of the lattice, and we investigate situations
where the bare value of $K$ does not exceed $J$. By using the non-Abelian
bosonization, we show that a coherent heavy-fermion phase can be stabilized,
when the bare value of $K$ exceeds the RKKY exchange (ruled by $T_A\simeq
J^2/t$). Conversely, when $K<T_A$, the low-energy fixed point is rather a
Tomonaga-Luttinger (TL) liquid.
