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

A Detailed Study of Bogomol'nyi Equations in Two-Dimensional Generalized Maxwell-Higgs Model Using \textit{On-Shell} Method