  The classical Chung-Feller theorem [2] tells us that the number of Dyck paths
of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on
$m$. L. Shapiro [9] found the Chung-Feller properties for the Motzkin paths.
Mohanty's book [5] devotes an entire section to exploring Chung-Feller theorem.
Many Chung-Feller theorems are consequences of the results in [5]. In this
paper, we consider the $(n,m)$-lattice paths. We study two parameters for an
$(n,m)$-lattice path: the non-positive length and the rightmost minimum length.
We obtain the Chung-Feller theorems of the $(n,m)$-lattice path on these two
parameters by bijection methods. We are more interested in the pointed
$(n,m)$-lattice paths. We investigate two parameters for an pointed
$(n,m)$-lattice path: the pointed non-positive length and the pointed rightmost
minimum length. We generalize the results in [5]. Using the main results in
this paper, we may find the Chung-Feller theorems of many different lattice
paths.
