Regex for positive or negative integer only with no look arounds

I would like to match integers only. No decimals and no leading 0s. Look arounds are not allowed. The best solution I had was ^\d+$ but it matches 0001 on https://regex101.com/.

Here are some test cases

1 
-1 
+1 
A1
)1 
0.1 
0001
1,000
1.1
0

This is what the result must be

1 #true
-1 #true
+1 #true
A1 #false
)1 #false
0.1 #false
0001 #false
1,000 #false
1.1 #false
0 #true

Edit

I also added the condition to match 0.