  We use a recent {\it on-shell} Bogomol'nyi method, developed
in~\cite{Atmaja:2014fha}, to construct Bogomol'nyi equations of the
two-dimensional generalized Maxwell-Higgs model~\cite{Bazeia:2012uc}. The
formalism can generate a large class of Bogomol'nyi equations parametrized by a
constant $C_0$. The resulting equations are classified into two types,
determined by $C_0=0$ and $C_0\neq0$. We identify that the ones obtained by
Bazeia {\it et al}~\cite{Bazeia:2012uc} are of the type $C_0=0$. We also
reveal, as in the case of ordinary vortex, that this theory does not admit
Bogomol'nyi equations in the Bogomol'nyi-Prasad-Sommerfield limit in its
spectrum. However, when the vacuum energy is lifted up by adding some constant
to the energy density then the existence of such equation is possible. Another
possibility whose energy is equal to the vacuum is also discussed in brief. As
a future of the \textit{on-shell} method, we find another new Bogomol'nyi
equations, for $C_0\neq0$, which are related to a non-trivial function defined
as a difference between energy density of potential term of the scalar field
and kinetic term of the gauge field.
