  The generalized sine-Gordon (sG) equation $u_{tx}=(1+\nu\partial_x^2)\sin\,u$
was derived as an integrable generalization of the sG equation. In a previous
paper (Matsuno Y 2010 J. Phys. A: Math. Theor. {\bf 43} 105204) which is
referred to as I, we developed a systematic method for solving the generalized
sG equation with $\nu=-1$. Here, we address the equation with $\nu=1$. By
solving the equation analytically, we find that the structure of solutions
differs substantially from that of the former equation. In particular, we show
that the equation exhibits kink and breather solutions and does not admit
multi-valued solutions like loop solitons as obtained in I. We also demonstrate
that the equation reduces to the short pulse and sG equations in appropriate
scaling limits. The limiting forms of the multisoliton solutions are also
presented. Last, we provide a recipe for deriving an infinite number of
conservation laws by using a novel B\"acklund transformation connecting
solutions of the sG and generalized sG equations.
