diff --git a/Combinatorial_map/include/CGAL/Creation_of_test_cases_for_paths.h b/Combinatorial_map/include/CGAL/Creation_of_test_cases_for_paths.h
index e77091d05bd..34baa5203d4 100644
--- a/Combinatorial_map/include/CGAL/Creation_of_test_cases_for_paths.h
+++ b/Combinatorial_map/include/CGAL/Creation_of_test_cases_for_paths.h
@@ -173,6 +173,24 @@ void generate_l_shape_case4(Path& p)
   extend_straight_negative(p, 2);
 }
 
+template<typename Path>
+void generate_l_shape_case5(Path& p)
+{ // (x -1 -2^t): here (-4 -1 -2^12)
+  p.clear();
+  p.push_back(p.get_map().darts().iterator_to(p.get_map().darts()[27]));
+  extend_uturn_negative(p, 1);
+  extend_straight_negative(p, 12);
+}
+
+template<typename Path>
+void generate_l_shape_case6(Path& p)
+{ // (x -2^t -1): here (-4 -2^12 -1)
+  p.clear();
+  p.push_back(p.get_map().darts().iterator_to(p.get_map().darts()[13]));
+  extend_straight_negative(p, 12);
+  extend_uturn_negative(p, 1);
+}
+
 template<typename Path>
 void generate_l_shape_case7(Path& p)
 { // (-3 -2^s -1 -2^t): here (-2^7 -1 -2^3 -3)
diff --git a/Combinatorial_map/include/CGAL/Path_on_surface.h b/Combinatorial_map/include/CGAL/Path_on_surface.h
index 6bdba66ac89..f53d0f8505e 100644
--- a/Combinatorial_map/include/CGAL/Path_on_surface.h
+++ b/Combinatorial_map/include/CGAL/Path_on_surface.h
@@ -509,6 +509,7 @@ public:
               }
             }
             while(next_turn==2);
+            begin=next_index(begin); // because we stopped on a dart s.t. next_turn!=2
           }
           // Here begin is the first dart of the path s.t. next_turn==-2
           // i.e. the previous turn != -2
diff --git a/Linear_cell_complex/examples/Linear_cell_complex/compute_example.cpp b/Linear_cell_complex/examples/Linear_cell_complex/compute_example.cpp
index bf78e3dd1a6..0635e17aaea 100644
--- a/Linear_cell_complex/examples/Linear_cell_complex/compute_example.cpp
+++ b/Linear_cell_complex/examples/Linear_cell_complex/compute_example.cpp
@@ -298,6 +298,7 @@ bool test_all_cases_spurs_and_bracket()
     res=false;
   }
 
+  // TODO Update spiral-squared.off to have only square
   lcc.clear();
   if (!CGAL::load_off(lcc, "./data/spiral-squared.off"))
   {
@@ -327,6 +328,7 @@ bool test_all_cases_spurs_and_bracket()
     res=false;
   }
 
+  // TODO Update loop-squared.off to have only square
   lcc.clear();
   if (!CGAL::load_off(lcc, "./data/loop-squared.off"))
   {
@@ -413,8 +415,7 @@ bool test_all_cases_l_shape()
 
   generate_l_shape_case4(path);
   // std::cout<<"Case 4: L-shape (-4 -2^7 -1 -2^3): "<<std::flush;
-  path.display_pos_and_neg_turns();std::cout<<std::endl;
-  push_l_shape(path, true, 1);
+  push_l_shape(path, false, 1);
   if (!path.same_turns("4 1 2 2 2 2 2 2 3 2 2 1"))
   {
     std::cout<<"[test_all_cases_l_shape case 4] ERROR: ";
@@ -423,6 +424,37 @@ bool test_all_cases_l_shape()
     res=false;
   }
 
+  lcc.clear();
+  if (!CGAL::load_off(lcc, "./data/cases5-6-right-shift-squared.off"))
+  {
+    std::cout<<"PROBLEM reading file ./data/cases5-6-right-shift-squared.off"<<std::endl;
+    exit(EXIT_FAILURE);
+  }
+  lcc.reverse_orientation();
+
+  generate_l_shape_case5(path);
+  // std::cout<<"Case 5: L-shape (-4 -1 -2^12): "<<std::flush;
+  path.display_pos_and_neg_turns();std::cout<<std::endl;
+  push_l_shape(path, true, 1);
+  if (!path.same_turns("4 2 2 2 2 2 2 2 2 2 2 2 2 1"))
+  {
+    std::cout<<"[test_all_cases_l_shape case 5] ERROR: ";
+    std::cout<<"we obtained "; path.display_pos_and_neg_turns();
+    std::cout<<" instead of (4 2 2 2 2 2 2 2 2 2 2 2 2 1)"<<std::endl;
+    res=false;
+  }
+
+  generate_l_shape_case6(path);
+  // std::cout<<"Case 6: L-shape (-4 -2^12 -1): "<<std::flush;
+  push_l_shape(path, false, 1);
+  if (!path.same_turns("4 1 2 2 2 2 2 2 2 2 2 2 2 2"))
+  {
+    std::cout<<"[test_all_cases_l_shape case 6] ERROR: ";
+    std::cout<<"we obtained "; path.display_pos_and_neg_turns();
+    std::cout<<" instead of (4 1 2 2 2 2 2 2 2 2 2 2 2 2)"<<std::endl;
+    res=false;
+  }
+
   // path.display_pos_and_neg_turns();std::cout<<std::endl;
   return res;
 }