  The $b$-family of Camassa-Holm ($b$-CH) equation is a one-parameter family of
PDEs, which includes the completely integrable Camassa-Holm and
Degasperis-Procesi equations but possesses different Hamiltonian structures.
Motivated by this, we study the existence and the orbital stability of the
smooth solitary wave solutions with nonzero constant background to the $b$-CH
equation for the special case $b=1$, whose the Hamiltonian structure is
different from that of $b\neq1$. We establish a connection between the
stability criterion for the solitary waves and the monotonicity of a singular
integral along the corresponding homoclinic orbits of the spatial ODEs. We
verify the latter analytically using the framework for the monotonicity of
period function of planar Hamiltonian systems, which shows that the smooth
solitary waves are orbitally stable. In addition, we find that the existence
and orbital stability results for $0<b<1$ are similar to that of $b>1$,
particularly the stability criteria are the same. Finally, combining with the
results for the case $b>1$, we conclude that the solitary waves to the $b$-CH
equation is structurally stable under the variation of %with respect to the
parameter $b>0$.
