diff --git a/parso/python/tokenize.py b/parso/python/tokenize.py
index 9bdf0c5..48d128b 100644
--- a/parso/python/tokenize.py
+++ b/parso/python/tokenize.py
@@ -54,8 +54,13 @@ if py_version >= 30:
     # Python 3 has str.isidentifier() to check if a char is a valid identifier
     is_identifier = str.isidentifier
 else:
-    namechars = string.ascii_letters + '_'
-    is_identifier = lambda s: s in namechars
+    # Python 2 doesn't, but it's not that important anymore and if you tokenize
+    # Python 2 code with this, it's still ok. It's just that parsing Python 3
+    # code with this function is not 100% correct.
+    # This just means that Python 2 code matches a few identifiers too much,
+    # but that doesn't really matter.
+    def is_identifier(s):
+        return True
 
 
 def group(*choices, **kwargs):
@@ -135,7 +140,11 @@ def _create_token_collection(version_info):
     Comment = r'#[^\r\n]*'
     # Python 2 is pretty much not working properly anymore, we just ignore
     # parsing unicode properly, which is fine, I guess.
-    if version_info[0] < 3 or sys.version_info[0] == 2:
+    if version_info[0] == 2:
+        Name = r'([A-Za-z_0-9]+)'
+    elif sys.version_info[0] == 2:
+        # Unfortunately the regex engine cannot deal with the regex below, so
+        # just use this one.
         Name = r'(\w+)'
     else:
         Name = u'([A-Za-z_0-9\u0080-' + MAX_UNICODE + ']+)'
diff --git a/test/test_diff_parser.py b/test/test_diff_parser.py
index 99c9327..04e1ca0 100644
--- a/test/test_diff_parser.py
+++ b/test/test_diff_parser.py
@@ -974,10 +974,12 @@ def test_random_unicode_characters(differ):
     Those issues were all found with the fuzzer.
     """
     differ.initialize('')
-    differ.parse(u'\x1dĔBϞɛˁşʑ˳˻ȣſéÎ\x90̕ȟòwʘ\x1dĔBϞɛˁşʑ˳˻ȣſéÎ', parsers=1, expect_error_leaves=True)
+    differ.parse(u'\x1dĔBϞɛˁşʑ˳˻ȣſéÎ\x90̕ȟòwʘ\x1dĔBϞɛˁşʑ˳˻ȣſéÎ', parsers=1,
+                 expect_error_leaves=True)
     differ.parse(u'\r\r', parsers=1)
     differ.parse(u"˟Ę\x05À\r   rúƣ@\x8a\x15r()\n", parsers=1, expect_error_leaves=True)
-    differ.parse(u'a\ntaǁ\rGĒōns__\n\nb', parsers=1)
+    differ.parse(u'a\ntaǁ\rGĒōns__\n\nb', parsers=1,
+                 expect_error_leaves=sys.version_info[0] == 2)
     s = '        if not (self, "_fi\x02\x0e\x08\n\nle"):'
     differ.parse(s, parsers=1, expect_error_leaves=True)
     differ.parse('')
diff --git a/test/test_tokenize.py b/test/test_tokenize.py
index 8d8f272..bf703dc 100644
--- a/test/test_tokenize.py
+++ b/test/test_tokenize.py
@@ -142,7 +142,7 @@ def test_identifier_contains_unicode():
     else:
         # Unicode tokens in Python 2 seem to be identified as operators.
         # They will be ignored in the parser, that's ok.
-        assert unicode_token[0] == OP
+        assert unicode_token[0] == ERRORTOKEN
 
 
 def test_quoted_strings():