Pick the solutions that satisfy certain conditions

Question

I have a large system of ODEs, and I find a large number of steady-state solutions. But many of them do not satisfy the condition that all the variables must have positive or zero values at the steady-state.

I like to:

1. retrieve only the solutions that satisfy the condition above;
2. sort the variables in a solution with respect to the variable (integer) subscript; and
3. put all the retrieved and sorted solutions in a table column-wise (each column should have a solution)

I have already done what I request above by using a mixture of If, AppendTo and Table commands but since the list is too long and it takes much time to achieve what I want, plus it looks ugly.

A minimal example of the desired output is:

TableForm[{{0.5, 2}, {1, 3}, {2, 1}}, 
 TableHeadings -> {{"\!\(\*SubscriptBox[\(x\), \(1\)]\)", 
    "\!\(\*SubscriptBox[\(x\), \(2\)]\)", 
    "\!\(\*SubscriptBox[\(x\), \(3\)]\)"}, {"solution 1", 
    "solution 2"}}]

EDIT 1

Below is a subset of solutions that directly come out of the model.

{
{Subscript[x, 1][t] -> -331, Subscript[x, 3][t] -> -1.8, 
  Subscript[x, 4][t] -> -32, Subscript[x, 2][t] -> -1.37, 
  Subscript[x, 6][t] -> -49.2, 
  Subscript[x, 5][t] -> -98},
{Subscript[x, 3][t] -> -1.84, 
  Subscript[x, 2][t] -> 1.6, Subscript[x, 4][t] -> 141.9, 
  Subscript[x, 1][t] -> 334.1, Subscript[x, 5][t] -> 0, 
  Subscript[x, 6][t] -> 10},
{Subscript[x, 6][t] -> 0, Subscript[x, 1][t] -> 35.13, 
  Subscript[x, 2][t] -> 0, Subscript[x, 4][t] -> 0, Subscript[x, 
  3][t] -> 2.92, Subscript[x, 5][t] -> 10.41},
{Subscript[x, 1][t] -> 11.17, 
  Subscript[x, 3][t] -> 14.7, Subscript[x, 4][t] -> 2.3, 
  Subscript[x, 2][t] -> -6.6, Subscript[x, 5][t] -> 3.3, 
  Subscript[x, 6][t] -> -7.13},
{Subscript[x, 1][t] -> 1, 
  Subscript[x, 2][t] -> 4, Subscript[x, 6][t] -> 0, 
  Subscript[x, 3][t] -> 3, Subscript[x, 5][t] -> 6, 
  Subscript[x, 4][t] -> 0}
}

The above set has 5 solutions, shown with {...} for each solution. As one may easily see, the variables in each solution are in somewhat mixed order with respect to variable subscripts. Here is a brief explanation for preparing the table.

1. The solutions should be sorted with respect to the variable subscripts, such as x1, x2, x3, etc.
2. Identify the solutions in which all variables have non-negative values, and place them in the early columns of the table according to the number of non-negative values of the variables. The more the non-negative values in a solution, the earlier the column position in the table.
3. Priority with respect to the column position should be given to those non-zero solutions with the highest number of non-negative values.
4. In case of multiple non-positive solutions, the ordering and column position of the solution set is not important.

EDIT 2

Here is the table of solutions I refer to in my comment.

The solution 2 should be in the first column of the table.

Answer 1

list = SortBy[#, #[[1, -1]] &] & /@ 
  SortBy[Select[solutions /. a_[t] :> a, And @@ NonNegative[#[[All, 2]]] &], 
   {Count[#[[All, -1]], _?Negative] &, -Count[#[[All, -1]], _?Positive] &}];
rlabels = ToString[#, StandardForm] & /@ list[[1, All, 1]];
clabels = "solution" <> ToString[#] & /@ Range[Length@list];
values = list[[All, All, 2]];
TableForm[Transpose @ values, TableHeadings -> {rlabels, clabels} , 
 TableAlignments -> Center]

If you want to keep all solutions: change list above to

list = SortBy[#, #[[1, -1]] &] & /@ 
   SortBy[solutions /. a_[t] :> a, 
    {Count[#[[All, -1]], _?Negative] &, -Count[#[[All, -1]], _?Positive] &}];