LIB "foliation.lib"; LIB "linalg.lib"; LIB "painleve-heun.lib";
ring r=(0,a,b,c,z),x,dp;
finalheun(g2,g3,*,*);

//list  1 I1 I1 I1 I9 
number g2=3*z*(9*z^3-8);
number g3=27*z^6-36*z^3+8;
g2=number(substpar(g2,z,z+1));
g3=number(substpar(g3,z,z+1));
((z+1)^3-1)/z; //(z2+3z+3)
 finalheun(g2,g3,3,3);

//list 2  I1-I1-I2-I8  With two exponent parameters a and c
number g2=3*(16*z^4-16*z^2+1);
number g3=64*z^6-96*z^4+30*z^2+1;

//list 3 I1-I2-I3-I6
number g2=12*(z^4-4*z^3+2*z+1);
number g3=4*(2*z^6-12*z^5+12*z^4+14*z^3+3*z^2+6*z+2);

//list  4 I1 I1 I5 I5 
number g2=3*(z^4-12*z^3+14*z^2+12*z+1);
number g3=z^6-18*z^5+75*z^4+75*z^2+18*z+1; //z^2-11*z-1
finalheun(g2,g3,-11,-1);


//list 5 I2-I2-I4-I4 with three exponent parameter a,b and c
number g2=12*(z^4-z^2+1);
number g3=4*(2*z^6-3*z^4-3*z^2+2);

//list  6 I3 I3 I3 I3 
number g2=3*(8*z^3+1);
number g3=(8*z^6+20*z^3-1);
g2=number(substpar(g2,z,z/(z-1)));
g3=number(substpar(g3,z,z/(z-1)));
poly aa=(g2^3-27*g3^2); //finalheun(g2,g3,-1,1/3);
//There are four singularities in $\C$, therefor we put Y=z, X=1
number g2=3*z*(8+z^3);
number g3=8+20*z^3-z^6;
g2=number(substpar(g2,z,z+1));
g3=number(substpar(g3,z,z+1));
(g2^3-27*g3^2)/((z2+3z+3)^3);
((1-(z+1)^3)/z; //(z2+3z+3)
finalheun(g2,g3,3,3);



//list  7 I1 I1 I8 II 
number g2=12*z*(z^3-6*z^2+15*z-12);
number g3=4*z*(2*z^5-18*z^4+72*z^3-144*z^2+135*z-27); //3z^2-14z+27
finalheun(g2,g3,-14/3,27/3);

//list 8 I1-I2-I7-II
number g2=12*z*(9*z^3+36*z^2+42*z+14);
number g3=12*z*(18*z^5+108*z^4+234*z^3+222*z^2+87*z+8);

//list 9 I1-I4-I5 II
number g2=3*(8*z-1)*(8*z^3+87*z^2+96*z-64);
number g3=(8*z-1)*(64*z^5+16*5*13*z^4+25*157*z^3+100*z^2+128*25*z-512);

//list 10 I2 I3 I5 II 
number g2=3*(z-3)*(81*z^3-9*z^2-53*z-27);
number g3=(z-3)*(3^6*z^5-3^5*5*z^4-2*3^3*5^2*z^3-350*z^2-3^3*5^2*z-243);

//list  11 I1 I1 I7 III 
number g2=12*z*(z^3+4*z^2+10*z+6);
number g3=4*z^2*(2*z^4+12*z^3+42*z^2+70*z+63); //4z^2+13z+32
finalheun(g2,g3,13/4,32/4);


//list 12 I1 I2 I6 III 
number g2=12*z*(z^3-6*z^2+9*z-3);
number g3=4*z^2*(2*z^4-18*z^3+54*z^2-63*z+27);

//list 13 I1-I3-I5-III;
number g2=75*(5*z-1)*(5*z^3+45*z^2+39*z-25);
number g3=25*(5*z-1)^2*(25*z^4+340*z^3+2*3*181*z^2+100*z+5^4);


// liste 14 I_2 I_3 I_4 III
number g2= 3*(z-1)*(16*z^3-3*z-1)  ;
number g3= (z-1)^2*(64*z^4+32*z^3+6*z^2+5*z+1)  ;
//number g2= 3*(1-z)*(16-3*z^2-z^3)  ;
//number g3= (1-z)^2*(64+32*z+6*z^2+5*z^3+z^4)  ;

// liste 15 I_1 I_1 I_6 IV
number g2= 3*z^2*(9*z^2-8)  ;
number g3= z^2*(27*z^4-36*z^2+8)  ;
//number g2= 3*(9-8*z^2)  ;
//number g3= 27-36*z^2+8*z^4  ;

// liste 16 I_1 I_2 I_5 IV
number g2= 12*z^2*(z^2+8*z+10) ;
number g3= 4*z^2*(2*z^4+24*z^3+78*z^2+56*z+27)  ;
//number g2= 12*(1+8*z+10*z^2) ;
//number g3= 4*(2+24*z+78*z^2+56*z^3+27*z^4)  ;

// liste 17 I_3 I_3 I_2 IV
number g2= 3*(z-1)^2*(9*z^2+14*z+9) ;
number g3= (z-1)^2*(27*z^4+36*z^3+2*z^2+36*z+27)  ;
//number g2= 3*(1-z)^2*(9+14*z+9*z^2) ;
//number g3= (1-z)^2*(27+36*z+2*z^2+36*z^3+27*z^4)  ;

// liste 18 I_1 I_7 II II
number g2= 3*(z^2-13*z+49)*(z^2-5*z+1) ;
number g3= (z^2-13*z+49)*(z^4-14*z^3+63*z^2-70*z-7)  ;// z2-13*z+49
//number g2= 3*(1-13*z+49*z^2)*(1-5*z+z^2) ;
//number g3= (1-13*z+49*z^2)*(1-14*z+63*z^2-70*z^3-7*z^4)  ;
finalheun(g2,g3,-13,49);



  
//list 19 I2-I6-II-II
number g2=3*(z-1)*(z+1)*(9*z^2-1);
number g3=(z-1)*(z+1)*(27*z^4-18*z^2-1);

// liste 20 I_4 I_4 II II
number g2=12*z*(2*z-1)*(z+4)  ;
number g3= 2*(2*z-1)*(z+4)*(z^4+4*z^3+8*z^1-4)  ;
g2=number(substpar(g2,z,(-z+1/2)/(z/4+1)));
g3=number(substpar(g3,z, (-z+1/2)/(z/4+1) ));
poly kk=(z^2+2*z-2);
kk=number(substpar(kk,z,(-z+1/2)/(z/4+1) ));
(g2^3-27*g3^2)/kk^4; //(z2-10z-2)
finalheun(g2,g3,-10,-2);
//number g2=12*z*(2-z)*(1+4*z)  ;
//number g3= 2*(2-z)*(1+4*z)*(1+4*z+8*z^3-4*z^4) ;
//g2=number(substpar(g2,z,(-z+2)/(4*z+1)));
//g3=number(substpar(g3,z, (-z+2)/(4*z+1)));
//poly kk=(1+2*z-2*z^2);
//kk=number(substpar(kk,z, (-z+2)/(4*z+1)));
//(g2^3-27*g3^2)/kk^3; //6z2+30z-3
  
// liste 21 I_1 I_6 II III
LIB "foliation.lib"; LIB "linalg.lib"; LIB "painleve-heun.lib"; LIB "complexroot.lib"; 
ring r=(0,a,b,c,z,tn),(x,y),dp;
number g2=2*z*(z-1)*(6*z^2+6*tn*z-(3+tn));
number g3=2*z^2*(z-1)*(4*z^3-2*(1-3*tn)*z^2-4*(2+tn)*z+(5-tn));
//We need only the SL form which is calculated inside the procedure finalheun. 
//Therefore, we modify finalheun by puting return(sss). Then we run  
number slform=number(finalheun(g2,g3));
ideal II=x^2+3;
II=std(II);
poly  deno=denominator(slform);
poly  nomi=deno*slform;
deno=substpar(deno,tn,x);
nomi=substpar(nomi,tn,x);
deno=reduce(deno,II);
nomi=reduce(nomi, II);
deno=subst(deno,x,tn);
nomi=subst(nomi,x,tn);
number deno1=number(deno);
number nomi1=number(nomi);
number  slform1=slform;
slform=nomi1/deno1;
	number sing1=(-2/7)*(3*tn+1);
	number sing2=0;
	number sing3=1;
        number aeins, azwei, adrei, avier, ele;
        number zmo=(39*z^2*tn+71*z^2+144*z*tn-160*z-48*tn-208)/(39*tn+71);  
        aeins=number(substpar(slform*zmo, z, sing1));

        deno=denominator(aeins);
        nomi=deno*aeins;
        nomi=reduce(subst(nomi,tn,x), II);
        deno=reduce(subst(deno,tn,x), II);
        deno1=number(subst(deno,x,tn));
        nomi1=number(subst(nomi,x,tn));
        aeins=nomi1/deno1; //aeins=(1/4*a^2-1/4*a-3/16);
        number the1h, the2h, the3h, the4einsh;
        the1h=4*aeins+1;
        number the1=wurzelnz(the1h);

	azwei=number(substpar(slform*(z-sing2)^2, z, sing2));
        adrei=number(substpar(slform*(z-sing3)^2, z, sing3));
        the2h=4*azwei+1;
	the3h=4*adrei+1;
        number the2=wurzelnz(the2h); 
	number the3=wurzelnz(the3h);


	slform=slform-aeins/zmo-azwei/(z-sing2)^2-adrei/(z-sing3)^2;
        number slform2=slform;
        
        number abc=slform*z*(z-1)*zmo;

        avier=number(diffpar(diffpar(abc,z),z))/2;
        the4einsh=4*avier+the1h+the2h+the3h-2;
        number the4=(wurzelnz(the4einsh)-1);
        slform=slform-avier/(z*(z-1));
        abc=slform*z*(z-1)*zmo;
        ele=diffpar(abc,z)/(sing1*(sing1-1));

        number Kpar=ele-(the1-1)*(sing1*the2+sing1*the3-2*sing1-the2+1)/(2*sing1^2-2*sing1);
        number th41=(-1/2)*(the1+the2+the3-1+the4); 
        number th42=(-1/2)*( the1+the2+the3-1-the4)+1;
        //Kpar is not reduced mod II
        poly AA=substpar(Kpar,a,x);
        ring rr=(0,tn),x,dp; minpoly=tn^2+3;
        poly AA=imap(r,AA);
        AA;
  
// liste 22 I_2 I_5 II III
number g2= 3*z*(2*z-27)*(2*z^2-35*z+140) ;
number g3= z*(2*z-27)^2*(2*z^3-39*z^2+222*z-250)  ;  
//number g2= 3*(2-27*z)*(2-35*z+140*z^2) ;
//number g3= (2-27*z)^2*(2-39*z+222*z^2-250*z^3)  ;
  
// liste 23 I_3 I_4 II III
number g2= 3*(z-1)*(z+27)*(16*z^2+80*z-243)  ;
 number g3= (z-1)^2*(z+27)*(64*z^3+2^5*43*z^2+2*3^5*z+3^9)  ;
//number g2= 3*(1-z)*(1+27*z)*(16+80*z-243*z^2)  ;
//number g3= (1-z)^2*(1+27*z)*(64+2^5*43*z+2*3^5*z^2+3^9*z^3)  ;
  
// liste 24 I_1 I_5 II IV
number g2= 3*z^2*(z-3)*(z+5) ;
number g3= z^2*(z-3)*(z^3+6*z^2-3*z-32)  ;
//number g2= 3*(1-3*z)*(1+5*z) ;
//number g3= (1-3*z)*(1+6*z-3*z^2-32*z^3)  ;
  

// liste 25 I_2 I_4 II IV
number g2= 36*z^2*(z-1)*(3*z-1) ;
number g3= 4*z^2*(z-1)*(54*z^3-54*z^2+9*z-1)  ;
//number g2= 36*(1-z)*(3-z) ;
//number g3= 4*(1-z)*(54-54*z+9*z^2-z^3)  ;
  
// liste 26 I_1 I_5 III III
number g2=3*(z^2+1)*(z^2+6*z+4);
number g3= (z^2+1)^2*(z^2+9*z+19);
g2=number(substpar(g2,z,z-11/2));
g3=number(substpar(g3,z,  z-11/2));
poly kk=z^2+1;
kk=number(substpar(kk,z, z-11/2) );
(g2^3-27*g3^2)/kk^3; //(z2-11z+125/4)
finalheun(g2,g3,-11,125/4);  

// liste 27 I_2 I_4 III III
number g2= 3*(z-1)*(z+1)*(4*z^2-1)  ;
number g3= (z-1)^2*(z+1)^2*(8*z^2+1);
//number g2= 3*(1-z)*(1+z)*(4-z^2)  ;
//number g3= (1-z)^2*(1+z)^2*(8+z^2);  
  

//list 28 I3 I3 III III
number g2=3*z*(z^2+6*z-3);
number g3=6*z^2*(z^2+3); //z^2-6*z-3; 
finalheun(g2,g3,-6,-3);

//list 29 I1 I4 III IV
number g2=12*z^2*(z-1)*(z+5);
number g3=4*z^2*(z-1)^2*(2*z^2+16*z+27);


//list  30 I2 I3 III IV
number g2=3*z^2*(z-1)*(9*z-5);
number g3=z^2*(z-1)^2*(27*z^2-9*z+2);

//list  31 I2 I2 IV IV
number g2=3*(z-1)^2*(z+1)^2;
number g3=(z-1)^2*(z+1)^2*(z^2+1);

//list  32 I2 IV III III
number g2=3*z^2*(z-1)*(z+1);
number g3=z^2*(z-1)^2*(z+1)^2;


//list  33 I3 III III III
number g2=3*z*(z^3-1);
number g3=(z^3-1)^2;
g2=number(substpar(g2,z,z+1));
g3=number(substpar(g3,z,  z+1)); \\((z+1)^3-1)/z=(z2+3z+3);
finalheun(g2,g3,3,3);


//list  34 I3 II III IV
number g2=3*z^2*(z-1)*(z+3);
number g3=z^2*(z-1)^2*(z+3)*(z+2);


//list 35 I4-IV-II-II
number g2=12*z^2*(z-1)*(z+1);
number g3=4*z^2*(z-1)*(z+1)*(2*z^2-1);


//list  36 
number g2=3*(z^2+2)*(z+5)*(z+1);
number g3=(z^2+2)^2*(z+5)*(z+4);
g2=number(substpar(g2,z,z-5));
g3=number(substpar(g3,z,  z-5));
poly kk=z^2+2;
kk=number(substpar(kk,z, z-5) );//(z2-10z+27)
finalheun(g2,g3,-10,27);


//list  37
number g2=3*z*(z^2+11*z+64)*(z+3);
number g3=z^2*(z^2+11*z+64)*(z^2+10*z+45);// z^+11*z+64
finalheun(g2,g3,11,64);

//list  38 
number g2=12*z*(z^3-1);
number g3=4*(z^3-1)*(2*z^3-1);
g2=number(substpar(g2,z,z+1));
g3=number(substpar(g3,z,  z+1));
poly kk=z^2+z+1;
kk=number(substpar(kk,z, z+1) );//(z2+3z+3)
finalheun(g2,g3,3,3);