############ Proc to extract one x;
#symm_sol = set of symmetry components from pdsolve in the form {eta_X=...,xh_t=...}
#N=Number of free functions/consts to be considered 
# spec: specification of free constants and functions which are to be taken non-zero, e.g., spec:={_C1=0,_C2=a,_F1=sin} 


gem_extract_symm:=proc(symm_sol, {N::integer:=100, spec:={}})
    local to_subs, ii, jj, const_list,func_list, spec_list, spec_listN, this_found;
    const_list:=linalg[vector](N);
    func_list:=linalg[vector](N);
    for ii from 1 to N do
       const_list[ii]:=cat('_C',ii);
       func_list[ii]:=cat('_F',ii);
    end do:    
    to_subs:={};
    spec_list:=convert(spec,list);
    spec_listN:=linalg[vectdim](spec_list);


    for ii from 1 to N do
       #constants       
       this_found:=0;
       for jj from 1 to spec_listN do
          if (has(spec_list[jj],const_list[ii])) then 
             this_found:=1;
             break;
          end if; 
       end do;
       if(this_found=0) then
          to_subs:=to_subs union {const_list[ii]=0};
       else
          to_subs:=to_subs union {spec_list[jj]};
       end if;  
       #functions       
       this_found:=0;
       for jj from 1 to spec_listN do
          if (has(spec_list[jj],func_list[ii])) then 
             this_found:=1;
             break;
          end if; 
       end do;
       if(this_found=0) then
          to_subs:=to_subs union {func_list[ii]=0};
       else
          to_subs:=to_subs union {spec_list[jj]};
       end if;
    end;  

    #print(to_subs);

    return simplify(eval(subs(to_subs,symm_sol)));
end:

# default global group parameter name: group_param_name='_a'
gem_global_group:=proc(symm_sol, {group_param_name:='_a'} )
    local ODE_IVP, curr_var, curr_RHS, subs_group_param_dep, g_p_name, ii, jj;

    ODE_IVP:={}:
    g_p_name:=eval(group_param_name);
    subs_group_param_dep:={seq(GEM_DEP_VARS[ii]=GEM_DEP_VARS[ii](g_p_name),ii=1..GEM_N_DEP_V), seq(GEM_INDEP_VARS[ii]=GEM_INDEP_VARS[ii](g_p_name),ii=1..GEM_N_INDEP_V)};

    #-Dependent
    for ii from 1 to GEM_N_DEP_V do
       curr_var:=GEM_DEP_VARS[ii]; #symbol name only
       curr_RHS:=subs(symm_sol,GEM_TVF_COMPONENTS:-dep_components[ii]);
       ODE_IVP:=ODE_IVP union {diff(GEM_DEP_VARS[ii](g_p_name),g_p_name)=eval(subs(subs_group_param_dep, curr_RHS)) };
       #Now ICs
       ODE_IVP:=ODE_IVP union {GEM_DEP_VARS[ii](0)=eval(cat(GEM_DEP_VARS[ii], '_0'))};
    end do:
    #-Independent
    for ii from 1 to GEM_N_INDEP_V do
       curr_var:=GEM_INDEP_VARS[ii]; #symbol name only
       curr_RHS:=subs(symm_sol,GEM_TVF_COMPONENTS:-ind_components[ii]);
       ODE_IVP:=ODE_IVP union {diff(GEM_INDEP_VARS[ii](g_p_name),g_p_name)=eval(subs(subs_group_param_dep, curr_RHS)) };
       #Now ICs
       ODE_IVP:=ODE_IVP union {GEM_INDEP_VARS[ii](0)=eval(cat(GEM_INDEP_VARS[ii], '_0'))};
    end do:

end:

# Procedure generating extra conditions that differ TVF components for arbitrary func/consts present in given DE system
#    and dep/indep variables in the system.
#
# parameter "dependence1" to be given in the form
# [ [[Q1],[d1,d2,d3]], [[Q2,Q3],[d1,d3,d7]], ...  ]
# where Qi are symbols (names) of independent or dependent variables or arbitrary constants/functions,  
#   whose Tangent Vector Field Components are to depend on d's, which are also
#   symbols (names) of independent or dependent variables or arbitrary constants/functions, 
#Example:symbols (names) of independent or dependent variables or arbitrary constants/functions,  
 

gem_generate_EquivTr_dependence:=proc(dependence1)
    local cond_EqTr, N_dependence_entries, non_dependence, kk, ii, jj, N1, curr_var;

    N_dependence_entries:=linalg[vectdim](dependence1);
    cond_EqTr:={}:
    for kk from 1 to N_dependence_entries do 
       # vars in dependence1[kk][1] will not depend on these: 
       non_dependence:=convert( convert(GEM_TVF_COMPONENTS:-dependence,set) minus convert(dependence1[kk][2],set)  ,list);
       N1:=linalg[vectdim](non_dependence); 

       #-Dependent
       for ii from 1 to GEM_N_DEP_V do
         curr_var:=GEM_DEP_VARS[ii]; #symbol name only
         if(has(dependence1[kk][1],curr_var)) then
             for jj from 1 to N1 do
               cond_EqTr:=cond_EqTr union { diff(eval( cat('eta_',curr_var))(GEM_TVF_COMPONENTS:-dependence[]), non_dependence[jj]) =0 };
             end do; 
         end if;
       end do:

       #-Independent
       for ii from 1 to GEM_N_INDEP_V do
         curr_var:=GEM_INDEP_VARS[ii]; #symbol name only
         if(has(dependence1[kk][1],curr_var)) then
             for jj from 1 to N1 do
               cond_EqTr:=cond_EqTr union { diff(eval( cat('xi_',curr_var))(GEM_TVF_COMPONENTS:-dependence[]), non_dependence[jj]) =0 };
             end do; 
         end if;
       end do:

    end do:

    return cond_EqTr; 

end: