diff --git a/Spatial_searching/include/CGAL/Fuzzy_sphere.h b/Spatial_searching/include/CGAL/Fuzzy_sphere.h
index 2ff94063b49..5953ac767a6 100644
--- a/Spatial_searching/include/CGAL/Fuzzy_sphere.h
+++ b/Spatial_searching/include/CGAL/Fuzzy_sphere.h
@@ -146,13 +146,17 @@ class Fuzzy_sphere< Search_traits_adapter<K,PM,Base_traits> >
 public:
   // constructors
   Fuzzy_sphere(const SearchTraits& traits_=SearchTraits()):Base(traits_){}
-  Fuzzy_sphere(const typename Base_traits::Point_d& center, FT radius, FT epsilon=FT(0),const SearchTraits& traits_=SearchTraits()) :
-    Base(center,radius,epsilon,traits_) {}
-  Fuzzy_sphere(const typename SearchTraits::Point_d& center, FT radius, FT epsilon=FT(0),
-               const SearchTraits& traits_=SearchTraits(),
+
+  // Constructor for any point type that is not `SearchTraits::Point_d`
+  template <typename Point> // boost::disable_if requires a template argument to work
+  Fuzzy_sphere(const Point& center, FT radius, FT epsilon=FT(0),const SearchTraits& traits_=SearchTraits(),
                typename boost::disable_if<
-                 boost::is_same<typename Base_traits::Point_d,
-                                typename SearchTraits::Point_d> >::type* = 0)
+               boost::is_same<typename SearchTraits::Point_d,
+               Point> >::type* = 0)
+    : Base(center,radius,epsilon,traits_) {}
+
+  Fuzzy_sphere(const typename SearchTraits::Point_d& center, FT radius, FT epsilon=FT(0),
+               const SearchTraits& traits_=SearchTraits())
     : Base(get(traits_.point_property_map(),center),radius,epsilon,traits_) {}
 };
 
diff --git a/Spatial_searching/include/CGAL/Search_traits_adapter.h b/Spatial_searching/include/CGAL/Search_traits_adapter.h
index f6d4eae7c6d..1fc87b33290 100644
--- a/Spatial_searching/include/CGAL/Search_traits_adapter.h
+++ b/Spatial_searching/include/CGAL/Search_traits_adapter.h
@@ -33,6 +33,7 @@
 
 #include <boost/mpl/has_xxx.hpp>
 #include <boost/type_traits/is_same.hpp>
+#include <boost/utility/enable_if.hpp>
 
 namespace CGAL{
 
@@ -111,11 +112,23 @@ public:
     // This is needed because of an undocumented requirement of 
     // Orthogonal_k_neighbor_search and Orthogonal_incremental_neighbor_search: 
     // Traits::Construct_cartesian_const_iterator should be callable 
-    // on the query point type
-    typename Base_traits::Cartesian_const_iterator_d operator()(const typename Base_traits::Point_d& p) const
+    // on the query point type. If the query point type is the same as
+    // Point_with_info, we disable it.
+
+    template <typename Point> // boost::disable_if requires a template argument to work
+    typename Base_traits::Cartesian_const_iterator_d operator()(const Point& p,
+                                                                typename boost::disable_if<
+                                                                boost::is_same<Point_with_info,
+                                                                Point> >::type* = 0
+      ) const
     { return Base::operator() (p); }
 
-    typename Base_traits::Cartesian_const_iterator_d operator()(const typename Base_traits::Point_d& p, int)  const
+    template <typename Point> // boost::disable_if requires a template argument to work
+    typename Base_traits::Cartesian_const_iterator_d operator()(const Point& p, int,
+                                                                typename boost::disable_if<
+                                                                boost::is_same<Point_with_info,
+                                                                Point> >::type* = 0
+      ) const
     { return Base::operator() (p,0); }
   };
   
diff --git a/Spatial_searching/test/Spatial_searching/Search_traits_adapter_with_identity_map.cpp b/Spatial_searching/test/Spatial_searching/Search_traits_adapter_with_identity_map.cpp
new file mode 100644
index 00000000000..00e66ca1d0f
--- /dev/null
+++ b/Spatial_searching/test/Spatial_searching/Search_traits_adapter_with_identity_map.cpp
@@ -0,0 +1,38 @@
+#include <CGAL/Simple_cartesian.h>
+#include <CGAL/Kd_tree.h>
+#include <CGAL/point_generators_2.h>
+#include <CGAL/algorithm.h>
+#include <CGAL/Fuzzy_sphere.h>
+#include <CGAL/Search_traits_2.h>
+#include <CGAL/property_map.h>
+
+typedef CGAL::Simple_cartesian<double> Kernel;
+typedef Kernel::Point_2 Point_2;
+typedef CGAL::Random_points_in_disc_2<Point_2> Random_points;
+
+typedef CGAL::Identity_property_map<Point_2> Point_map;
+typedef CGAL::Search_traits_2<Kernel> Search_base;
+typedef CGAL::Search_traits_adapter<Point_2, Point_map, Search_base> Search_traits;
+typedef CGAL::Fuzzy_sphere<Search_traits> Fuzzy_circle;
+typedef CGAL::Kd_tree<Search_traits> Tree;
+
+int main ()
+{
+  Random_points rdpts;
+  std::vector<Point_2> pts;
+  for (std::size_t i = 0; i < 50; ++ i)
+    pts.push_back (*(rdpts ++));
+
+  Tree tree (pts.begin(), pts.end());
+    
+  Point_2 center(0., 0.);
+  Fuzzy_circle circle (center, 0.5);
+  std::vector<Point_2> result;
+  tree.search(std::back_inserter(result), circle);
+  std::cout << "The points in the fuzzy circle centered at (0., 0.) ";
+  std::cout << "with fuzzy radius (0.5) are: " << std::endl;
+  for (std::size_t i = 0; i < result.size(); ++ i)
+    std::cout << " * " << result[i] << std::endl;
+  
+  return 0;
+}