A Homothetic Utility is where $$ \forall x,y, \forall a \in \mathbb{R}_+: \ u(ax,ay)=au(x,y) $$ (or its monotonic transformation).
A linear Homothetic utility is defined as $$ \forall x,y, \forall a \in \mathbb{R}_+: \ u(ax+b,ay+c)=au(x+b,y+c) $$ where $b,c$ are constants.
This preference has very similar property as the homothetic preference. In fact, if we simply translate the coordinate system in the direction of (b,c), then the preference becomes homothetic.
Are there any works covering this property? I've checked a lot of theory papers in homothetic preference but found no luck.
Homothetic Preferences by James DOW· and Sergio Ribeiro da Costa WERLANG
Homothetic and weakly homothetic preferences by J.C. Candeal, E. Indurain
Linear-homothetic preferences