logical deduction - Special 5x5 Sudoku-like table

This is a special 5x5 Sudoku-like table.
Each row, column, and diagonal contains digits from 1 to 5 (including diagonals obtained by wrapping around the edges).
To filter duplicates (by rotation, reflection, permutations), I put numbers in ascending order (1,2,3,4,5) to the first row of the tables.

We can create another table like this, by logical deduction.

Find another solution, by filling the blanks squares!

Answer

Here is another solution:

1 2 3 4 5
3 4 5 1 2
5 1 2 3 4
2 3 4 5 1
4 5 1 2 3

(There's not much to say about the method of solution. I tried putting something other than 5 in the (2,2) position; there was only one option; there were only two options for the next thing to the right, one of which quickly turned out impossible; after that it was just a matter of filling in the only possible number at each step. So this is the only solution that has 12345 along the top and doesn't have 5 in position (2,2). Perhaps there are other solutions with 5 at (2,2); I haven't checked; but I suspect not.)