Using same exogenous variable in mean and variance

[anedo]

Hi - it is great to have this forum

Here's my question, I'd be very happy if I could get some ideas:

I am doing a simple GARCH(1,1) model with 1 exogenous variable whose effect
I want to test on both, the mean and the variance. Is it correct to use this same
variable simultaneously in both equations?

Secondly, is it correct to use this same exogenous variable both, in M-GARCH, and again, the variance equation?

Could you please interpret what it means?

Thanks a lot!