# n = 5 in the following condition:
# (i) When f ramifies of order 2 above any one of {0,1,infty} with exponent difference <> odd integer/2.

Deg2_5:= proc(L::set,x,K)    

# L is the singularity structure of input differential operator, 
# i.e. a set of lists [pi,ei] where pi is an irreducible poly in x 
# and ei its corresponding exponent difference.
# K is the base field.

 local L1,a2,i,i1,j,j1,k,k1,P,Ps,Es,N,C1,C2,e0,e1,ei,Base_Field,f,f1,F,Field_F,cand,c,c_val;
 
if nargs = 2 then 
  Base_Field:= indets([args], {nonreal, RootOf, radical});
  return procname(args, Base_Field);
else Base_Field:= K;
fi;

# we are looking for the rational f which produces 5 non 
# removable singularities from 0,1,infinity.
# So, we must have at least 3 irreducible polys and the polys may have at most degree 2.
# at least one of them must have degree 1:
  
L1:= L;
for i in L1 do if points(i[1],x) <> 1 and points(i[1],x) <> 2 then 
    return "Wrong input"; fi;
od;
if nops(L1) < 3 or add(points(i[1],x),i= L1) <> 5 then return "Not in Case Deg2_5"; fi;

Ps:=[seq(i[1],i=L1)];  Es:= [seq(i[2],i=L1)];  N:= {seq(i,i=1..nops(L1))};

# Compute f now: let's put the ramified point at the root of f:
# f:= k1*(x-a0)^2/(x^2+b1*x+b0);  1-f:= k2*(x^2+c1*x+c0)/(x^2+b1*x+b0);
# where other polynomials have non repeated roots.

cand:= {};
  for i in N do
     if points(Ps[i],x) <> 1 then next; fi;
       if type(Es[i],integer) then e0:= normal(Es[i]/2); 
       else e0:= {normal(Es[i]/2), normal((Es[i]+1)/2)};
       fi;
        C1:= N minus {i};
    f:= c*(Ps[i])^2;
      for j in C1 do         
         ei:= Es[j]; C2:= C1 minus {j}; a2:= points(Ps[j],x);
         f1:= f/Ps[j];
         if a2 = 1 then 
           for j1 in C2 while a2 <> 2 do if points(Ps[j1],x) = 1 and (type(evala(Es[j1]-ei),integer) or type(evala(Es[j1]+ei),integer)) then
           a2:= a2+ points(Ps[j1],x);
           C2:= C2 minus {j1};
           f1:= f1/Ps[j1]; fi; od;
           if a2 <> 2 then next; fi;
           fi;
           e1:= Es[C2[1]];
           if nops(C2) = 2 then
              if not type(evala(Es[C2[2]]-e1),integer) and not type(evala(Es[C2[2]]+e1),integer) then next; fi;
           fi;
           P:= mul(Ps[k], k = C2);
           c_val:= {solve({coeffs(rem(numer(1-f1),P,x),x)})};
            if c_val <> {} and c_val <> {{}} then
              for k1 in c_val do F:= eval(f1,c=rhs(op(k1)));
                if max(degree(numer(F),x),degree(denom(F),x)) = 2 and type(rem(P, evala(Factor(numer(1-F))),x),numeric) then 
                Field_F:= indets(F, {RootOf, radical}) union `if`(has(F,I),{I},{});
                  if nops(Field_F minus Base_Field) <> 0 then next; fi;  
                    cand:= cand union {seq([factor(F),[k2,e1,ei]], k2 = e0)};
                fi;
              od;
           fi;       
       od; 
  od;
 cand;                   
end:



points:= proc(f,x)
if f = 1 then 1 else degree(f,x); fi;
end: