Basic question about dummy variables for breakpoint treatment

I am studying basic Econometrics and trying to understand how to deal with breakpoints using dummy variables.

I found 3 significant break-points in my data (using 5% confidence) with the Chow Breakpoint test.

I then estimated a model discarding all data before the first break, which produced a new model with no break points but with very low R^2 (17%).

So I created a dummy variable taking 0 before the first break and 1 after the break (until the end of data), and interacted it with my independent variable so:

y = c(1) + c(2)*X + c(3)*D + c(4)*X*D

D being the dummy variable. This produced a model with a much better R^2 (34%) [this is financial returns data]

Next, I found that if I interact the dummy with another variable the adjusted R^2 goes to 38%, but only the interaction of the dummy to this new variable is significant, the new variable by itself is not, so I removed it.

y = c(1) + c(2)*X + c(3)*D + c(4)*X*D + c(5)*Z*D

Z being the new variable.

I have two problems here:

First: I don't know if I can include just the interaction of the dummy with the new variable or should also include the "standalone" variable (even when p-value is not significant).

Second: I'm trying to run a break point test again to see if I still have the other two break points in this new model but I run into an error which says "specification leads to singular matrix in at least one sub-sample".

I've searched online and it looks like this is due to the dummy being constant after the break point. But then, how do I know if I still have break points when using this dummy variable to account for the first break point I found?