  This paper aims at solving a one-dimensional backward stochastic differential
equation (BSDE for short) with only integrable parameters. We first establish
the existence of a minimal $L^1$ solution for the BSDE when the generator $g$
is stronger continuous in $(y,z)$ and monotonic in $y$ as well as it has a
general growth in $y$ and a sublinear growth in $z$. Particularly, the $g$ may
be not uniformly continuous in $z$. Then, we put forward and prove a comparison
theorem and a Levi type theorem on the minimal $L^1$ solutions. A Lebesgue type
theorem on $L^1$ solutions is also obtained. Furthermore, we investigate the
same problem in the case that $g$ may be discontinuous in $y$. Finally, we
prove a general comparison theorem on $L^1$ solutions when $g$ is weakly
monotonic in $y$ and uniformly continuous in $z$ as well as it has a stronger
sublinear growth in $z$. As a byproduct, we also obtain a general existence and
unique theorem on $L^1$ solutions. Our results extend some known works.
