Linear Homothetic Utility
The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, $x$ and $y$, this takes the form:
$$
u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}.
$$
This is a Cobb-Douglas type of utility function where $a$ and $b$ are subsistence levels, i.e. you need to consume at least $a$ from $x$ and $b$ from $y$ to survive. It is the utility function that leads to the Linear expenditure system.
To see that it is linear homothetic notice that:
$$
\begin{align*}
u(\beta \tilde x + a, \beta \tilde y + b) &amp;= (\beta \tilde x + a - a)^\alpha (\beta \tilde y + b - b)^{1-\alpha},\\
&amp;=\beta (\tilde x)^\alpha (\tilde y)^{1-\alpha},\\
&amp;=\beta ((\tilde x + a) - a)^\alpha ((\tilde y + b) - b)^{1-\alpha},\\
&amp;= \beta u(\tilde x + a, \tilde y + b).
\end{align*}
$$
It could be that there is more work on utility functions with subsistence levels that lead to other preferences that are also linear homothetic.
For example you could define a CES utility function with subsistence levels:
$$
u(x,y) = (\alpha_x(x -a)^\sigma + \alpha_y(y-b)^\sigma)^{1/\sigma}
$$
This will also satisfy linear homogeneity. This paper of Baumgärtner, Drupp &amp; Quaas does something like this.
In general, if you take any homothetic utility function $u(x,y)$ then the modified 'subsistence-augmented' function:
$$
\tilde u(x, y) \equiv u(x - a, y - b),
$$
will be linear homothetic.