@Mariusz Iwaniuk I can't vote up posts, because I don't have enough reputation. I will vote up your posts once I have enough reputation.

@Mariusz Iwaniuk Thank you, it's nice to see a solution depending on four variables.

@vv You say that it is not possible to express the general solution in terms of real functions. I believe you are right, but I am not sure what that means for my search for solutions. But clearly that's not a Maple problem anymore. Thank you too.
 

@Mariusz Iwaniuk Thank you, this is already nice. But the system is so heavily underdetermined that one would expect to be able to freely choose three functions in four variables. (This statement is to be taken with a grain of salt and comes from exterior differential systems calculus)

If you assume that all functions only depend on the variables lambda, mu, and l, one may choose f3 and f4 arbitrarily and finds f1 and f2 to be uniquely determined. But it would be nice to see a generic solution depending on four variables.