A generalized SHGI integrable hierarchy and its expanding integrable model

An anti-symmetric loop algebra $overline{A}_2$ is constructed. It follows that an integrable system is obtained by use of Tu's scheme. The eminent feature of this integrable system is that it is reduced to a generalized Schr?dinger equation, the well-known heat-conduction equation and a Gerdjkov－Ivanov (GI) equation. Therefore, we call it a generalized SHGI hierarchy. Next, a new high-dimensional subalgebra $tilde{G}$ of the loop algebra $tilde{A}_2$is constructed. As its application, a new expanding integrable system with six potential functions is engendered.