Gen:=proc(L)
  local Sing, i, g, S; S:={};
  Sing:=singularity(L) minus removable(L);
  for i in Sing do 
        g:=DEtools[gen_exp](L,T,x=i); g:=[seq(op(i[1..-2]),i=g)]; S:=S union {[op(g),i]};
  od; S;
end:

sing2:=proc(f)   #find roots of f,1-f and 1/f.
   if degree(numer(f))>degree(denom(f))
      then [solve(numer(f),x),solve(f-1,x),infinity]
      elif degree(numer(f))<degree(denom(f))
           then [infinity,solve(f-1,x),solve(denom(f),x)]
           elif lcoeff(denom(f))=lcoeff(numer(f))
                then [solve(numer(f),x),infinity,solve(denom(f),x)]
                else [solve(numer(f),x),solve(f-1,x),solve(denom(f),x)]
   fi; 
end:





# case1 deals with case of {rootof(u,1-u,1/u)}={rootof(v,1-v,1/v)}  
# if u=v then our algorithm EQN(F1(a,b1,b2,c,u,v)) will return error
Case1:=proc(L,u,v)
 local GG,l1,e2,j,l2,g,e,C,B1,B2,A,L1,Lp,a1,a2,r,i,l,e1,L2,s,S;S:={};
 GG:=Gen(L); l1:=sing2(u); l2:=sing2(v); L1:=EQN(F1(a,b1,b2,c,u,v),x);
 if l1[1]=l2[2] then
       for g in GG do
          if g[-1]=l1[1] then e:=MatchExp1(g[1],g[2],g[3],0,b2-c+1,c-a-b2); break;fi; 
       od; 
       elif l1[1]=l2[3] then
            for g in GG do
               if g[-1]=l1[1] then e:=MatchExp1(g[1],g[2],g[3],a,b2,1+2*b2-c); break;fi; 
            od;               
       else for g in GG do
               if g[-1]=l1[1] then e1:=MatchExp1(g[1],g[2],g[3],0,1-c,2-c); break;fi; 
            od;
  fi;  
  if assigned(e) then
    for i in e do
      l:=solve(i,{a,c}); A:=op(op(l)[1])[2]; C:=op(op(l)[2])[2];
      L2:=subs(a=A,c=C,L1);
      L2:=collect(normal(L2),{Dx,Dx^2,Dx^3},factor);
      a1:=coeff(L,Dx^2); a2:=coeff(L2,Dx^2);
      r:=1/3*(a1-a2);
      Lp:=L2-DEtools[symmetric_product](L,Dx-r);
      Lp:=collect(Lp,Dx,normal):
      s:=solve({coeffs(numer(Lp),{Dx,x})},{b1,b2}); 
      if s<>NULL then
      B1:=op(op(s)[1])[2]; B2:=op(op(s)[2])[2]; C:=subs(b2=B2,C);A:=subs(s,A);r:=subs(s,r); S:=S union {[A,B1,B2,C,u,v,r]}; fi;
    od;
    else for i in e1 do
      C:=solve(i,c);
      for g in GG do
               if g[-1]=l1[2] then e2:=MatchExp1(g[1],g[2],g[3],a,b2,C-a-b1+b2); break;fi; 
      od;
      for j in e2 do
          l:=solve(j,{b1,b2}); B1:=op(op(l)[1])[2]; B2:=op(op(l)[2])[2];
          L2:=subs(b1=B1,b2=B2,c=C,L1);
          L2:=collect(normal(L2),{Dx,Dx^2,Dx^3},factor);
          a1:=coeff(L,Dx^2); a2:=coeff(L2,Dx^2);
          r:=1/3*(a1-a2);
          Lp:=L2-DEtools[symmetric_product](L,Dx-r);
          Lp:=collect(Lp,Dx,normal):
          A:=solve({coeffs(numer(Lp),{Dx,x})},a); 
          if A<>NULL then
          B1:=subs(a=A,B1); B2:=subs(a=A,B2); S:=S union {[A,B1,B2,C,u,v,r]}; fi;
      od;
      od;          
  fi; S;
end: