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On a real hypersurface 
$M$ in a complex two-plane Grassmannian 
${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ we have the Lie derivation 
$\mathcal{L}$ and a differential operator of order one associated with the generalized Tanaka–Webster connection 
${{\widehat{\mathcal{L}}}^{\left( k \right)}}$. We give a classification of real hypersurfaces $M$ on ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ satisfying 
$\widehat{\mathcal{L}}_{\xi }^{\left( k \right)}\,S\,=\,{{\mathcal{L}}_{\xi }}S$, where 
$\xi$ is the Reeb vector field on $M$ and 
$s$ the Ricci tensor of $M$.
There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians 
${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$. Among them, Suh classified Hopf hypersurfaces in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ with Reeb parallel Ricci tensor in Levi–Civita connection. In this paper, we introduce the notion of generalized Tanaka–Webster 
$\left( \text{GTW} \right)$ Reeb parallel Ricci tensor for Hopf hypersurfaces in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$. Next, we give a complete classification of Hopf hypersurfaces in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ with 
$\text{GTW}$ Reeb parallel Ricci tensor.
