Question: Issue with PDEtools:-Solve

I have a complicated set of first-order differential equations, which Maple seems capable of solving. So far so good. The obtained solution is fed into another complicated set of first-order differential equations, which, again, Maple seems capable of solving.

But this final (combined) solution cannot possibly be the complete one as it does not contain a specific (sub)solution which DOES satisfy both sets of equations (checked in Maple by evaluating these equations for that specific solution). And there is no warning raised concerning any solutions that may have been lost.

The problem seems to be that the integrational constants associated with solving (in turn) the two sets of equations become messed up together, or erroneously 'reused', in some mysterious way. For if I rename the integration constants of the first solution before feeding it into the second set of equations, then this specific subsolution IS contained in the final solution.

Therefore the following questions:

1.) Could there possibly be some erroneous 'reuse' of integration constants going on? It seems to me that in solving the second set of equations, PDEtools:-Solve would have to make sure that it numbers any new integration constants in such a way that there would be no conflict with the ones obtained in solving the first set of equations.

2.) Is there some (global) variable that determines what letter is assigned for the integration constants? I could use any such to easily switch names between the first and second solving.

PS: I think, it makes little sense to upload any code/worksheet here.