diff --git a/NEWS b/NEWS
index b51db0cf1..1f2303187 100644
--- a/NEWS
+++ b/NEWS
@@ -298,6 +298,11 @@ New in spot 2.9.8.dev (not yet released)
   - tgba_determinize() could create parity automata using more colors
     than necessary.  (Related to issue #298)
 
+  - There were two corners cases under which sat_minimize() would
+    return the original transition-based automaton when asked to
+    produce state-based output without changing the acceptance
+    condition.
+
   Deprecation notices:
 
   - twa_graph::defrag_states() and digraph::defrag_states() now take
diff --git a/spot/twaalgos/dtbasat.hh b/spot/twaalgos/dtbasat.hh
index a2a480566..79f096a1f 100644
--- a/spot/twaalgos/dtbasat.hh
+++ b/spot/twaalgos/dtbasat.hh
@@ -1,5 +1,5 @@
 // -*- coding: utf-8 -*-
-// Copyright (C) 2013, 2014, 2015 Laboratoire de Recherche et
+// Copyright (C) 2013-2015, 2021 Laboratoire de Recherche et
 // Développement de l'Epita.
 //
 // This file is part of Spot, a model checking library.
@@ -38,7 +38,7 @@ namespace spot
   /// turning the TBA into a BA.
   ///
   /// If no equivalent deterministic TBA with \a target_state_number
-  /// states is found, a null pointer
+  /// states is found, this returns a null pointer.
   SPOT_API twa_graph_ptr
   dtba_sat_synthetize(const const_twa_graph_ptr& a,
                       int target_state_number,
diff --git a/spot/twaalgos/dtwasat.cc b/spot/twaalgos/dtwasat.cc
index e3c625ba4..670a9ffc8 100644
--- a/spot/twaalgos/dtwasat.cc
+++ b/spot/twaalgos/dtwasat.cc
@@ -1472,16 +1472,22 @@ namespace spot
     // returned in the case of a WDBA.
     complete_here(a);
 
+    bool orig_is_valid = false;
+
     if (states == -1 && max_states == -1)
       {
         if (state_based)
           max_states = sbacc(a)->num_states();
         else
           max_states = a->num_states();
-        // If we have not user-supplied acceptance, the input
-        // automaton is a valid one, so we start the search with one
-        // less state.
-        max_states -= !user_supplied_acc;
+        // If we have no user-supplied acceptance, and we are not
+        // guessing state-based upperbound, the input automaton is a
+        // valid one, so we start the search with one less state.
+        if (!user_supplied_acc && (!state_based || a->prop_state_acc()))
+          {
+            --max_states;
+            orig_is_valid = true;
+          }
       }
 
 
@@ -1525,7 +1531,7 @@ namespace spot
               (a, state_based, sat_langmap, max_states);
         }
 
-        if (!a && !user_supplied_acc)
+        if (!a && orig_is_valid)
           a = orig;
       }
     else
diff --git a/tests/python/satmin.ipynb b/tests/python/satmin.ipynb
index 4cec11ad3..ff14b3195 100644
--- a/tests/python/satmin.ipynb
+++ b/tests/python/satmin.ipynb
@@ -1925,7 +1925,7 @@
    "source": [
     "How did the procedure look for a complete automaton of size 5 when the input had only 2 states?  It's because the input uses transition-based acceptance: to estimate an upper bound of the size of the state-based output, the `sat_minimize()` procedure converted its transition-based input to state-based acceptance (using the `spot.sbacc()` function) and counted the number of states in the result.\n",
     "\n",
-    "Such an estimate is not necessarily correct of we request a different acceptance condition.  In that case We can actually change the upper-bound using `max_states`.  Below we additionally demonstrate the use of the `colored` option, to request all transition to belong to exactly one set, as customary in parity automata."
+    "Such an estimate is not necessarily correct if we request a different acceptance condition.  In that case we can actually change the upper-bound using `max_states`.  Below we additionally demonstrate the use of the `colored` option, to request all transitions to belong to exactly one set, as customary in parity automata."
    ]
   },
   {
@@ -4253,7 +4253,7 @@
    "source": [
     "### `states`\n",
     "\n",
-    "Sometimes we do not want a minimization loop, we just want to generate an equivalent automaton with a given number of states.  In that case, we use the `states` option.  However there is no constraint that all state should be reachable, so in the end, you could end with an automaton with fewer states than requested."
+    "Sometimes we do not want a minimization loop, we just want to generate an equivalent automaton with a given number of states.  In that case, we use the `states` option.  However there is no constraint that all states should be reachable, so in the end, you could end with an automaton with fewer states than requested."
    ]
   },
   {
@@ -4588,7 +4588,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.1+"
+   "version": "3.9.2"
   }
  },
  "nbformat": 4,
diff --git a/tests/python/satmin.py b/tests/python/satmin.py
index 156b27ea8..2d28dd405 100644
--- a/tests/python/satmin.py
+++ b/tests/python/satmin.py
@@ -1,5 +1,5 @@
 # -*- mode: python; coding: utf-8 -*-
-# Copyright (C) 2015, 2020  Laboratoire de Recherche et Développement
+# Copyright (C) 2015, 2020, 2021  Laboratoire de Recherche et Développement
 # de l'Epita
 #
 # This file is part of Spot, a model checking library.
@@ -237,3 +237,13 @@ min4 = spot.sat_minimize(aut, acc='parity max odd 3',
                          colored=True, sat_naive=True)
 assert min4.num_sets() == 3
 assert min4.num_states() == 2
+
+
+aut = spot.translate('GFa')
+assert aut.num_sets() == 1
+assert aut.num_states() == 1
+assert aut.is_deterministic()
+out = spot.sat_minimize(aut, state_based=True)
+assert out.num_states() == 2
+out = spot.sat_minimize(aut, state_based=True, max_states=1)
+assert out is None