  A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is
a lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XOY-plane
that consists of up steps $\mathbf{u}=(1, 1)$, down steps $\mathbf{d}=(1, -1)$,
horizontal steps $\mathbf{h}=(1, 0)$ and vertical steps $\mathbf{v}=(0, -1)$.
An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the
$\mathbf{u}$-steps, $\mathbf{h}$-steps, $\mathbf{v}$-steps and
$\mathbf{d}$-steps are weighted respectively by $1, a, b$ and $c$. In this
paper, we first give bijections between the set of $\mathbf{uvu}$-avoiding
$(a,b,b^2)$-G-Motzkin paths of length $n$ and the set of $(a,b)$-Schr\"{o}der
paths as well as the set of $(a+b,b)$-Dyck paths of length $2n$, between the
set of $\{\mathbf{uvu, uu}\}$-avoiding $(a,b,b^2)$-G-Motzkin paths of length
$n$ and the set of $(a+b,ab)$-Motzkin paths of length $n$, between the set of
$\{\mathbf{uvu,uu}\}$-avoiding $(a,b,b^2)$-G-Motzkin paths of length $n+1$
beginning with an $\mathbf{h}$-step weighted by $a$ and the set of $(a,b)$-Dyck
paths of length $2n+2$. In the last section, we focus on the enumeration of
statistics "number of $\mathbf{z}$-steps" for $\mathbf{z}\in \{\mathbf{u},
\mathbf{h}, \mathbf{v}, \mathbf{d}\}$ and "number of points" at given level in
$\mathbf{uvu}$-avoiding G-Motzkin paths. These counting results are linked with
Riordan arrays.
