need help in the questions in the pic

I'm running a regression with a continuous response and various explanatory variables. some quantitative and some qualitative. I save the residuals and plot them against the explanatory variables and I see some patterns that suggest there is unequal variance or heteroscedasticity in relation to some of the explanatory variables. Now I know that regression books recommend finding a better model or applying a variance stabilizing transformation to eliminate this violation of the usual assumptions. But I think my model is just fine as it is and what I'd like to do is model the variance right along with the expectation of the response, all in the same model, simultaneously. The pattern of variances is substantively interesting! Could you help me? In particular, I have three questions: 1. What are the technical names or "keywords" for this sort of model, if it exists? 2. Could you write down a statistical model for this problem (one or two lines) and explain something about how it works? I need enough information to tell whether your model will help me answer the substantive questions that relate variances to explanatory variables. In particular, I need to model a group difference in variances and as well as patterns where the variances seem to be functions of the explanatory variables. 3. If we can identify a promising model, how will I be able to estimate it? Can you recommend computer software