  In this paper, we consider the knot complement problem for not
null-homologous knots in homology lens spaces. Let $M$ be a homology lens space
with $H_1(M; \mathbb{Z}) \cong \mathbb{Z}_p$ and $K$ a not null-homologous knot
in $M$. We show that $K$ is determined by its complement if $M$ is
non-hyperbolic, $K$ is hyperbolic, and $p$ is a prime more than 7, or, if $M$
is actually a lens space $L(p,q)$ and $K$ represents a generator of
$H_1(L(p,q))$.
