@mmcdara first answer of your question is: for case sigma<0 it is mention in below of picture also i will upload other in here.
secondly, i don't know how you did really is emazing code and about equation you did a good job,

but mistak is in begin of equation  when you replace (s_ode_1 := eval(s_ode, S)) this is mistak you should take derivative S then replace G'' so you get w' and then again take derivative from S' again replace G'' then put do eval (S,S',S'') which is w,w',w'' put in equation then do other stuff i explain in mp file better please check also i will upload Image of other case of getting system equation in the paper also if you send me email i will send you whole paper which i can't upload paper here for you to see
check.mw

sigma>0

sigma=0 is

@janhardo i don't have problem in algebra i have problem with code if are you expert go a head we wait for your solution prove yourself  i did a trail i failed so go ahead yo do .

@mmcdara  i am sorry for spending your time and i am very glade to achieve this information , i want  solve whole problem by achieve system of equation and factoring them but i prefer to use other formula like in image if posible can you check how i can get thus system i did a lot trail and i will put my maple file here , explanation is like this replace eq(6) in eq(5) and use eq(7) when we get G'' replace eq(7) in eq(6) each time when you get derivative replace eq(7) then put all derivative in our ode equation then factoring all term in below of image write then we have a system of algebra equation then substitute must be satisfy .

aproach_of_GGGG.mw

@mmcdara  it will be totlay correct  if instead of sigma we write sigma=A[1]^2-A[2]^2 will be true but don't know where sigma come maybe it is integral constant of solution after simplify and factoring.
also we have three type for lambda >0 and lambda=0 if you do them too it will be a big help for me becuase it is your talent i can't go jump in your code i try to change to lambda<0 but  i failed, also i want solve the whole paper of yesterday when you done and told me trick of getting (G'/G)^2 =....., if you help me at there it will be a big help becuase i stuck at factoring and isolating (1/G)^i and (1/G)^i*diff(G(xi),xi)/G also instead each (G'/G)^2 we must do substitute for each case seperatly all the same if we do one case other will be easy,i hope help me at there really i am stcuk at there,  also i am intresting in your coding is like you know what you doing to be a perfect, your help is emazing for my job.

this is other case at same paper

@acer it is totaly different from this question maybe picture are  same but demand of the question is different

@mmcdara  i must write the equation on top  picture but i write the other ode equation if you want you can change the ode to that one on top with first term s[1]w''-(mu[3]+s[1]... =0 like that or if you want i can send you pdf file if i have your email for more detail

@mmcdara i can't see any file with my maple version and i fixed your version code by simplify denominator of eq_9 i get the exact equation .

if you have time watch this file and if have any idea tell me i will do i need find thus equation in tope for my new ODE

i hope you help me this time too just for  factoring

find_parameter.mw

@mmcdara first sorry for my bad for sending a lot of question, secondly your code is totally true i can see that but the code for version of maple is not work some time have the same problem  i don't know why ...
nevermind if you can solve the equation and what code where be used for substitute if you have time it will be so good i am stuck really i know what they done in paper but for my new paper i must have a good code and if have any question i am here thank you so much

@janhardo this is a paper,

Various optical solutions for time-fractional Fokas system arises in monomode optical fibers

[full PDF link removed by moderator]

@mmcdara i watch in your uploade is correct but when i run have problem is like that

@mmcdara 6533 the denominator of eq 9 is not the same of equation  11 and 13 have the same code but outcome is different  just (1/G)^2 have a little problem

@mmcdara  i just intrested in your code my code are a little bit ugly
can you substitute eq(6) in eq(5) and also when take derivative of eq(6) must be replace G''=-sigma*G(eta)+v eq(6) and when we substite must be factorize and equal coificent to zero but we must substitute our equation like (eq9),eq(11)and eq(13)
by collecting term is  hard for substitute i don't know there is another better way for factoring and replacing

you already did the hard part of this if you do this part i will be very greatfull

@mmcdara it is wonderfull and awesome and with your explanation i hope  undrestand full subject

@Thomas Richard  it is  PDE by a lot of step i reach to this, this solution is satify for ode wothout 1/n but when we substitute to pde for test is equal to zero or not i must add power 1/n becuase the transformation i did a lot of this kind equation but i don't know what is mistak in here

@janhardo this is emazing, thank you for making rich the subject