@mehdibaghaee: Your quadruple add seems ok to me. And in view of the previously mentioned problem with using SumOverRepeatedIndices, this would also be the way I would proceed. Your approach also has the advantage that the values of II and JJ can differ, a thing that would seem to be impossible in the setup I gave.

PS: By the way, I suddenly realize that your summations start at 0, not 1 as in the setup I provided.

@Ronan: Happy to hear that $include works for you.

Concerning indentation: Depending on what you precisely have in mind (I suppose it is option 2 below):
1.) You can indent as many lines as you like in one go by simply marking them and pressing the tab-button.
2.) You can have the next line be indented as the former one by choosing the 'simple'-option under Utilities, Global options, Automatic Indentation.

Concerning output from read: What I had in mind was first and foremost outputting to screen statements that would have been outputted as well if the worksheet had been written and executed normally from within Maple. To display the contents of a module, I do not really know. If this is not done with the module written and executed normally from within Maple, using a semicolon, then surely it will not work when executed using the read statement. However, you can always get your hands on, say, the exports of a module by writing exports(modulename). But perhaps that is not what you had in mind.

@tomleslie: Quite right.

@tomleslie: You are absolute right, I had overlooked item 1 altogether, as well as part of item 2. Your solution seems to generate the same list everytime, though.

@vv: Thanks for your suggestion. I will have a look at it tomorrow; it is time to call it a day :-). I am afraid, though, that I will no longer have any need for it now that I have realized my mistake as outlined in the 'ups' to John May. Sorry for the inconvenience.

Update: Morning here in Denmark, back at the keyboard. I have checked your suggestion, and it works out nicely. And I understand why. Although I stated the opposite yesterday, there is actually more than a fair chance that I will use your solution, the reason being the following: I want to relate the square root of a product of some complex numbers to the product of their square roots. As far as I understand, this will be well-defined if no product of any pair of these numbers 'crosses' the branch cut. With the set of complex numbers that I have, I can guarantee that with a branch cut out along the negative imaginary axis.

@John May: The actual branch cut is of course a matter of choice, no problem there. But the formula you give is not well-defined for values x lying on the negative real axis, because the logarithm itself has a branch cut there.

Ups, consulting https://en.wikipedia.org/wiki/Square_root, I suddenly realize that the argument can actually lie on the branch cut itself. I had fooled myself into believing it was not allowed to do so, i.e., that phi (in polar coordinates) should belong to ]-Pi,+Pi[, and not, as is actually the case, to ]-Pi,+Pi]. Somewhat embarrasing, but at least I am a little bit wiser :-). As you probably can tell, I have not had much experience with branch cuts.

@Gillee: "You could use DeleteColumn() and DeleteRow() after determining if the sum of the elements in the column or row add up to zero." Only if all elements, in some row or column, are guaranteed either all positive, or all negative; otherwise a sum could be zero even if not all the elements summed are.

@ecterrab: No offense taken. I understand your point of view. The burden of proof lies with me. I will return, here or in a new thread depending on the time span, if and when I succeed in providing something concrete.

@Preben Alsholm: Thanks for your example. I am, however, not sure whether the problem I experience with my equations arise in the same way as the 'no good'- example of yours. At present the problem is still quite opaque to me.

@Rouben Rostamian: I agree.

It seems I will have to trim my system nonetheless, to either persuade myself that there actually is no problem, or to persuade Edgardo that there is. If this trimming is at all possible, it will probably have to wait a while as 1.) I have other things to attend to for the moment, and 2.) the 'renaming trick' seems to bypass the issue for the moment. If I do manage to trim the system, and convince myself there (still) is a problem, then I will report back, either here or in a new thread, depending on the time span.

@tomleslie: I think that you have caught the essence of the problem. With that simple and specific example at hand, I think I will stop trying to trim my system. Thanks.

@ecterrab: I am still trying to trim the system, while maintaining the issue. Although the work is not done yet, I will shut down my computer for today.

@ecterrab: Thanks for weighing in.

I believe it makes little sense to upload my worksheet, as it uses commands and quantities from various packages I have created over the years. So it would neither execute by itself, nor be understandable for an outsider. So in order to reproduce the problem, I will have to 'flesh it out'. At present, though, it is completely opaque to me where to 'trim' the system to make it a stand-alone one. If that changes, I will return.

@tomleslie: I agree, the issue would disappear if the two systems were solved simultaneously (at least if the indeterminates of that combined system are chosen not to possibly conflict with integration constants, etc., that may arise when solving the system). Perhaps I should consider doing that. But, on the other hand, I would like to break down the underlying physical in the two-stage manner I have presently adopted. So the jury is still out :-).

Thanks for the explanation on the variable separation constants. Although I have, of course, learned about separation of variables in relation to integration of differential equations, it would seem that it is so long ago, I had somehow forgot about it, ups! At least, I did not make the connection here in Maple.