@Carl Love Many thanks! After your message, I started using assuming instead of assume, and as you said, it's definitely easier to work with.

@dharr Many many thanks. Perhaps you could help me to understand why adding the following assumptions makes it worse.

assume(beta,'real',beta>0);
assume(sigma,'real',sigma>0);
assume(rho,'real',rho>0);

@aroche Sure, here they are.  I have both Maple 2024 and 2025 and the results are the same.

eqqq := map(simplify,{sqrt(beta*rho*xi[8]^2 - beta*xi[8]^2)/xi[8] = sqrt(beta*rho - beta), (-rho + 1 + sqrt(beta*rho*xi[8]^2 - beta*xi[8]^2))/(xi[3]*xi[8] - 1) = sqrt(beta*rho - beta), -(-rho*xi[3]*xi[8] + sqrt(beta*rho*xi[8]^2 - beta*xi[8]^2) + xi[3]*xi[8])/(xi[8]*(xi[3]*xi[8] - 1)) = rho - 1});

sol:=solve(eqqq,{xi[8],xi[3]});

simplify(subs(sol,eqqq));

evalf(subs({sigma=10,beta=8/3,rho=28},%));

 @dharr I was surprised that solve and SolveTools:-PolynomialSystem returned a different number of solutions, so I ran a small experiment. I took all combinations of six unknowns and used both solve and SolveTools:-PolynomialSystem to compute the solutions: solve returned 2,170 solutions, while SolveTools:-PolynomialSystem returned only 21 (after doing my best to eliminate duplicates and common solutions). The difference is significant.

Update: I have to check the solutions further.

@dharr I can't use solve or SolveTools:-PolynomialSystem as you did.   They are always divided into two sets: one containing \alpha_{9,*,*} and the other containing \alpha_{14,*,*}. When a solution is found using one of these sets, it always depends on the other set or includes a free parameter. I found only one solution when using all the unknowns \alpha_{9,*,*}.

The answer to your question is yes, they are different solutions.

@dharr Many thanks for pointing this out. The only issue is that, unfortunately, I am trying to find all solutions without any prior knowledge of what they are. So it seems that I will have to check a certain number of combinations of unknowns.

@Carl Love Many thanks.   I don't have a large list right now but I will in a near future.   The last two procedures will be of a great help once I have large lists to analyze. 

@Carl Love Many thanks.   Your explanation was very clear.   

@Carl Love Many thanks.  Sorry but I still have some questions on how the procedures work. 

1) Nterms – it should return the number of terms (monomials) in a polynomial regardless of how the polynomial is presented, correct?

evalb(Nterms(F1)=Nterms(simplify(F1)))

returns false. 

2) How can I include (x4, x3, x2, x1) as an argument to be passed to Key? I work with all kinds of polynomial variables, as you probably remember.

@ecterrab Thanks.  I maybe completely mistaken but DifferentialAlgebra tools needs the definition of a differential ring using derivations and blocks which is not the case in my example.  Even using R:=DifferentialRing(derivations=[t],blocks=[x1,x2,x3,x4]) or R:=DifferentialRing(derivations=[t],blocks=[x]) (and x1=x[t],x2=x[t,t],...) does not result in the order in my example.  

@Carl Love Many thanks.  Yes, simply "degree in x4 (regardless of the presence of other variables)".  I knew it that you would come up a single line to solve the problem while I would spend several lines to do the same. 

One thing is missing: the number of terms in the polynomial. The polynomials in x3​ (F4, F5) have the same degree (2), but the number of terms is 6 and 5, respectively, in your answer. There are also two polynomials in x4 with degree 1, but fortunately, they are in the correct order. This means a check for the number of terms should be performed whenever two or more polynomials belong to the same class (like x3​, for example).

The final order is: [F2,F3,F1,F5,F4,F6]

Obs.: Could you explain why subs is necessary and how key selects the correct degree to consider?

@Carl Love It does once I removed all the aliases from the author's source code.  If I don't remove them, SaveMyProcs saves the procedure without the "d_" (Example: it saves dmcs and not d_cms).  No follow-up question and sorry for the confusion. 

@Carl Love Many thanks. I tried using "suffixed" (not together with "And") before asking the question to see which procedures would be saved. I noticed that procedures such as "dmcs" and "depend" (that do not start with "d_") were also listed. Is there a way to save only the procedures that start with "d_" (including the underscore)?

I found out what was going on.   The author used the command alias (dmcs = d_cms).

@Carl Love Many thanks.   How to add a condition?  I need to save all procedures that start with "d_". 

@dharr It seems that I'll go through all of them one by one.  

Command completion om mac - Settings - interface.   

Many thanks.