@Carl Love Many thanks.   I hope my days of sending silly questions will be over soon (I have been bothering you since 2016!).  

@acer Thank you.   Please don't apologize. The answer was spot on in the first post. 

@acer  Many thanks.  I didn't imagine that it would be that easy.  Many thanks.  Since you've mentioned the possibility of forming both in a single call, please show how it should be done.    

@Thomas Richard Many thanks.   It seems that the only improvement was from 2017 to 2018 and no change after that.

@Joe Riel Thanks.  

The output would be 

coef:=[c1 c2 c3 1 c4]

and if the order in L is changed to

L := [c2*x^2 + c3*x*y + z + c1*x, c4*z^2]:

the output is

coef:=[c2 c3 1 c1 c4];

Somehow I think Rootof messes up the order.  

@Joe Riel Thanks.   

Issuing the command 

map2(map2,coeff,model7,vars);

gives the following result

[[-RootOf(64*_Z^3+80*_Z^2+1104*_Z+561)-5/4, alpha[1, 2], 0], [z*alpha[2, 6], RootOf(64*_Z^3+80*_Z^2+1104*_Z+561), x*alpha[2, 6]], [-17*y/alpha[2, 6]+2*z, -17*x/alpha[2, 6], 2*x-1]]

which is clearly wrong.  

@acer I believe both will work. Many thanks.

@tomleslie Many thanks.   So simple and I could not come up with such a  solution.  I still have a long way ...

@Carl Love Many thanks. 

@Kitonum Many thanks. IsolateNonlinearTerms does not seem to work when the coefficients are not numeric. (Curiosity since the polynomials in the question have terms with coefficients equal to 1).

Example:

w:=[[F*a, -a*x, -y, -z], [-b*x*z, x*y, G, -y], [b*x*y, x*z, -z]]

then when the function is applied the result is 

[F*a, -a*x, -b*x*z, x*y, b*x*y, x*z], [-y, -z, G, -y, -z]

@Carl Love Many thanks for helping me again. A comment on 3*x is given in another reply.

@dharr Many thanks.  In all my lists of polynomials from which the nonlinear terms have been extracted the coefficients of all terms are equal to one. That is the reason why 3*x is not an issue.   Anyways, the two solutions are far better than mine. I have a lot to learn. Thank you both ever so much.  

@acer I am not sure if I can give a good answer as requested.  I haven't said that the solution is optimal or best but simply it is one that fits my needs.  Yes, if just one variable is excluded, all the better.   I really don't know how to break ties.  

I don't know if the following definition will good enough for you but I will give a shot.  Definition: given a set of nonlinear equations, how to find the maximal set of unconflicted variables (and therefore a solution) and how to reduce the inconsistency to a minimal?  Sorry if I can do a better job.   Thanks for your patience.  

@acer Many thanks.   In the example given above, only alpha[2,3] and alpha [3,5] cannot be found (inconsistency). The remaining alphas can be found.  What I need is exactly that: find what is possible to find and show me the equations that cannot be solved.  (alpha[1,2]=-193/100,alpha[2.7]=-74/125,alpha[2,2]=477/1000,alpha[2,1]=139/100,alpha[1,4]=629/500,alpha[2,5]=1093/500).

@Carl Love Hi, Carl.  It would certainly.   That will make my life easier.  I have so many of these set of equations that anything to "shorten" them out would help. Many thanks.