Obj function yielding independent goods demand functions
So if we take a simple example with 2 goods and consider the consumer's problem:

$L=u(x,y)+\lambda(I-p_xx-p_yy)$

This yields the ratio of FOC:
$u_1/u_2=p_x/p_y$

Now plugging these guys into the budget:

$I=p_xu_1^{-1}(u_2p_x/p_y)+p_yy$

A sufficient condition is $u_1^{-1}$ be hod -1 to make $p_x$ drop out. But I don't think it's necessary. 

Now I'm not exactly sure what that means (the inverse of the marginal utility of a good is homogeneous of degree -1), but a great thing to think about.