diff --git a/NEWS b/NEWS
index 1119bdbc5..f9d52fff2 100644
--- a/NEWS
+++ b/NEWS
@@ -26,11 +26,18 @@ New in spot 1.1.4a (not relased)
       the man page of ltlcross for details.
 
     - There is a new command, named dstar2tgba, that converts a
-      deterministic Rabin or Streett automaton as output by
-      ltl2dstar into a TGBA, BA or Monitor.   When a deterministic
-      Rabin automaton is realizable by a deterministic Büchi automaton,
-      the conversion preserve determinism.   (This is not implemented
-      for Streett.)
+      deterministic Rabin or Streett automaton (expressed in the
+      output format of ltl2dstar) into a TGBA, BA or Monitor.
+
+      In the case of Rabin acceptance, the conversion will output a
+      deterministic Büchi automaton if one such automaton exist.  Even
+      if no such automaton exists, the conversion will actually
+      preserves the determinism of any SCC that can be kept
+      deterministic.
+
+      In the case of Streett acceptance, the conversion produces
+      non-deterministic Büchi automata with Generalized acceptance.
+      These are then degeneralized if requested.
 
     - The %S escape sequence used by ltl2tgba --stats to display the
       number of SCCs in the output automaton has been renamed to %c.
diff --git a/src/dstarparse/Makefile.am b/src/dstarparse/Makefile.am
index a09e59e24..a50feabd0 100644
--- a/src/dstarparse/Makefile.am
+++ b/src/dstarparse/Makefile.am
@@ -52,7 +52,7 @@ EXTRA_DIST = $(DSTARPARSE_YY)
 
 libdstarparse_la_SOURCES = \
   fmterror.cc \
-  dra2dba.cc \
+  dra2ba.cc \
   dstar2tgba.cc \
   nra2nba.cc \
   nsa2tgba.cc \
diff --git a/src/dstarparse/dra2dba.cc b/src/dstarparse/dra2ba.cc
similarity index 71%
rename from src/dstarparse/dra2dba.cc
rename to src/dstarparse/dra2ba.cc
index c10039f94..f77a2cb13 100644
--- a/src/dstarparse/dra2dba.cc
+++ b/src/dstarparse/dra2ba.cc
@@ -54,6 +54,14 @@ namespace spot
   {
     typedef std::list<const state*> state_list;
 
+    // The function that takes aut and dra is first arguments are
+    // either called on the main automaton, in which case dra->aut ==
+    // aut, or it is called on a sub-automaton in which case aut is a
+    // masked version of dra->aut.  So we should always explore the
+    // automaton aut, but because the state of aut are states of
+    // dra->aut, we can use dra->aut to get labels, and dra->acccs to
+    // retrive acceptances.
+
     static bool
     filter_states(const tgba* aut,
 		  const dstar_aut* dra,
@@ -212,17 +220,23 @@ namespace spot
 
 
 
-    class dra_to_dba_worker: public tgba_reachable_iterator_depth_first
+    class dra_to_ba_worker: public tgba_reachable_iterator_depth_first
     {
     public:
-      dra_to_dba_worker(const tgba_explicit_number* a, const state_set& final):
-	tgba_reachable_iterator_depth_first(a),
+      dra_to_ba_worker(const dstar_aut* a,
+		       const state_set& final,
+		       const scc_map& sm,
+		       const std::vector<bool>& realizable):
+	tgba_reachable_iterator_depth_first(a->aut),
 	in_(a),
-	out_(new tgba_explicit_number(a->get_dict())),
-	final_(final)
+	out_(new tgba_explicit_number(a->aut->get_dict())),
+	final_(final),
+	num_states_(a->aut->num_states()),
+	sm_(sm),
+	realizable_(realizable)
       {
-	bdd_dict* bd = a->get_dict();
-	bd->register_all_variables_of(a, out_);
+	bdd_dict* bd = a->aut->get_dict();
+	bd->register_all_variables_of(a->aut, out_);
 
 	// Invent a new acceptance set for the degeneralized automaton.
 	int accvar =
@@ -243,36 +257,104 @@ namespace spot
 		   const state* sout, int,
 		   const tgba_succ_iterator* si)
       {
-	int in = in_->get_label(sin);
-	int out = in_->get_label(sout);
+	int in = in_->aut->get_label(sin);
+	int out = in_->aut->get_label(sout);
+	unsigned in_scc = sm_.scc_of_state(sin);
 
 	typedef state_explicit_number::transition trans;
 	trans* t = out_->create_transition(in, out);
 	bdd cond = t->condition = si->current_condition();
 
-	if (final_.find(sin) != final_.end())
-	  t->acceptance_conditions = acc_;
+	if (realizable_[in_scc])
+	  {
+	    if (final_.find(sin) != final_.end())
+	      t->acceptance_conditions = acc_;
+	  }
+	else if (sm_.scc_of_state(sout) == in_scc)
+	  {
+	    // Create one clone of the SCC per accepting pair,
+	    // removing states from the Ui part of the (Li, Ui) pairs.
+	    // (Or the Ei part of Löding's (Ei, Fi) pairs.)
+	    bitvect& l = in_->accsets->at(2 * in);
+	    bitvect& u = in_->accsets->at(2 * in + 1);
+	    for (size_t i = 0; i < in_->accpair_count; ++i)
+	      {
+		int shift = num_states_ * (i + 1);
+		// In the Ui set. (Löding's Ei set.)
+		if (!u.get(i))
+		  {
+		    // Transition t1 is a non-deterministic jump
+		    // from the original automaton to the i-th clone.
+		    //
+		    // Transition t2 constructs the clone.
+		    //
+		    // Löding creates transition t1 regardless of the
+		    // acceptance set.  We restrict it to the non-Li
+		    // states.  Both his definition and this
+		    // implementation create more transitions than needed:
+		    // we do not need more than one transition per
+		    // accepting cycle.
+		    trans* t1 = out_->create_transition(in, out + shift);
+		    t1->condition = cond;
+
+		    trans* t2 = out_->create_transition(in + shift,
+							out + shift);
+		    t2->condition = cond;
+		    // In the Li set. (Löding's Fi set.)
+		    if (l.get(i))
+		      t2->acceptance_conditions = acc_;
+		  }
+	      }
+	  }
       }
 
     protected:
-      const tgba_explicit_number* in_;
+      const dstar_aut* in_;
       tgba_explicit_number* out_;
-      const tgba* d_;
       const state_set& final_;
+      size_t num_states_;
       bdd acc_;
+      const scc_map& sm_;
+      const std::vector<bool>& realizable_;
     };
 
   }
 
 
-  tgba* dra_to_dba(const dstar_aut* dra)
+  tgba* dra_to_ba(const dstar_aut* dra, bool* dba)
   {
     assert(dra->type == Rabin);
 
     state_list final;
     state_list nonfinal;
-    if (!filter_scc(dra->aut, dra, final, nonfinal))
-      return 0;
+
+    // Iterate over all non-trivial SCCs.
+    scc_map sm(dra->aut);
+    sm.build_map();
+    unsigned scc_max = sm.scc_count();
+
+    bool dba_realizable = true;
+    std::vector<bool> realizable(scc_max);
+
+    for (unsigned scc = 0; scc < scc_max; ++scc)
+      {
+	if (sm.trivial(scc))
+	  {
+	    realizable[scc] = true;
+	    continue;
+	  }
+
+	// Get the list of states of that SCC
+	const std::list<const state*>& sl = sm.states_of(scc);
+	assert(!sl.empty());
+	bool scc_realizable = filter_states(dra->aut, dra, sl, final, nonfinal);
+	dba_realizable &= scc_realizable;
+	realizable[scc] = scc_realizable;
+	//std::cerr << "realizable[" << scc << "] = " << scc_realizable << "\n";
+      }
+
+    if (dba)
+      *dba = dba_realizable;
 
     // for (state_list::const_iterator i = final.begin();
     // 	 i != final.end(); ++i)
@@ -282,7 +364,7 @@ namespace spot
     //   std::cerr << dra->aut->get_label(*i) << " is non-final\n";
 
     state_set fs(final.begin(), final.end());
-    dra_to_dba_worker w(dra->aut, fs);
+    dra_to_ba_worker w(dra, fs, sm, realizable);
     w.run();
     tgba_explicit_number* res1 = w.result();
     tgba* res2 = scc_filter_states(res1);
diff --git a/src/dstarparse/dstar2tgba.cc b/src/dstarparse/dstar2tgba.cc
index 3af64598f..a82006802 100644
--- a/src/dstarparse/dstar2tgba.cc
+++ b/src/dstarparse/dstar2tgba.cc
@@ -26,14 +26,9 @@ namespace spot
     switch (daut->type)
       {
       case spot::Rabin:
-	{
-	  tgba* res = dra_to_dba(daut);
-	  if (res)
-	    return res;
-	  return nra_to_nba(daut);
-	}
+	return dra_to_ba(daut);
       case spot::Streett:
-	return spot::nsa_to_tgba(daut);
+	return nsa_to_tgba(daut);
       }
     assert(!"unreachable code");
     return 0;
diff --git a/src/dstarparse/nra2nba.cc b/src/dstarparse/nra2nba.cc
index accaf1790..f4472dc5c 100644
--- a/src/dstarparse/nra2nba.cc
+++ b/src/dstarparse/nra2nba.cc
@@ -37,7 +37,7 @@ namespace spot
     public:
       // AUT is the automate we iterate on, while A is the automaton
       // we read the acceptance conditions from.  Separating the two
-      // makes its possible to mask AUT, as needed in dra_to_dba().
+      // makes its possible to mask AUT, as needed in dra_to_ba().
       nra_to_nba_worker(const dstar_aut* a, const tgba* aut):
 	tgba_reachable_iterator_depth_first(aut),
 	out_(new tgba_explicit_number(aut->get_dict())),
diff --git a/src/dstarparse/public.hh b/src/dstarparse/public.hh
index d635d05ca..7e79f2852 100644
--- a/src/dstarparse/public.hh
+++ b/src/dstarparse/public.hh
@@ -117,16 +117,23 @@ namespace spot
   nra_to_nba(const dstar_aut* nra, const state_set* ignore);
 
   /// \brief Convert a deterministic Rabin automaton into a
-  /// deterministic Büchi automaton when possible.
+  /// Büchi automaton, deterministic when possible.
   ///
   /// See "Deterministic ω-automata vis-a-vis Deterministic Büchi
   /// Automata", S. Krishnan, A. Puri, and R. Brayton (ISAAC'94) for
-  /// more details.
+  /// more details about a DRA->DBA construction.
   ///
-  /// If the DRA is DBA-realizable, return that DBA.  Otherwise,
-  /// return NULL.
+  /// We essentially apply this method SCC-wise.  If an SCC is
+  /// DBA-realizable, we duplicate it in the output, fixing just
+  /// the acceptance states.   If an SCC is not DBA-realizable,
+  /// then we apply the more usual conversion from Rabin to NBA
+  /// for this part.
+  ///
+  /// If the optional \a dba_output argument is non-null, the
+  /// pointed Boolean will be updated to indicate whether the
+  /// returned Büchi automaton is deterministic.
   SPOT_API tgba*
-  dra_to_dba(const dstar_aut* dra);
+  dra_to_ba(const dstar_aut* dra, bool* dba_output = 0);
 
   /// \brief Convert a non-deterministic Streett automaton into a
   /// non-deterministic tgba.
@@ -135,8 +142,7 @@ namespace spot
 
   /// \brief Convert a Rabin or Streett automaton into a TGBA.
   ///
-  /// For DRA, this function uses dra_to_dba() when possible, or fall
-  /// back to nra_to_nba().
+  /// This function calls dra_to_ba() or nsa_to_tgba().
   SPOT_API tgba*
   dstar_to_tgba(const dstar_aut* dstar);