Index: src/where.c
==================================================================
--- src/where.c
+++ src/where.c
@@ -2479,14 +2479,16 @@
       pNew->wsFlags |= WHERE_COLUMN_EQ;
       assert( saved_nEq==pNew->u.btree.nEq );
       if( iCol==XN_ROWID 
        || (iCol>=0 && nInMul==0 && saved_nEq==pProbe->nKeyCol-1)
       ){
-        if( iCol>=0 && pProbe->uniqNotNull==0 ){
-          pNew->wsFlags |= WHERE_UNQ_WANTED;
-        }else{
+        if( iCol==XN_ROWID || pProbe->uniqNotNull 
+         || (pProbe->nKeyCol==1 && pProbe->onError && eOp==WO_EQ) 
+        ){
           pNew->wsFlags |= WHERE_ONEROW;
+        }else{
+          pNew->wsFlags |= WHERE_UNQ_WANTED;
         }
       }
     }else if( eOp & WO_ISNULL ){
       pNew->wsFlags |= WHERE_COLUMN_NULL;
     }else if( eOp & (WO_GT|WO_GE) ){

Index: test/join2.test
==================================================================
--- test/join2.test
+++ test/join2.test
@@ -153,23 +153,68 @@
 
 do_execsql_test 4.1.4 {
   SELECT v1, v3 FROM c1 LEFT JOIN c2 LEFT JOIN c3 ON (c3.k=v1+1);
 } {2 v3 2 v3 1112 {} 1112 {}}
 
-do_eqp_test 4.2.1 {
+do_eqp_test 4.1.5 {
   SELECT v1, v3 FROM c1 LEFT JOIN c2 ON (c2.k=v1) LEFT JOIN c3 ON (c3.k=v2);
 } {
   0 0 0 {SCAN TABLE c1} 
   0 1 1 {SEARCH TABLE c2 USING INTEGER PRIMARY KEY (rowid=?)}
   0 2 2 {SEARCH TABLE c3 USING INTEGER PRIMARY KEY (rowid=?)}
 }
-do_eqp_test 4.2.2 {
+do_eqp_test 4.1.6 {
   SELECT v1, v3 FROM c1 LEFT JOIN c2 ON (c2.k=v1) LEFT JOIN c3 ON (c3.k=v1+1);
 } {
   0 0 0 {SCAN TABLE c1} 
   0 1 2 {SEARCH TABLE c3 USING INTEGER PRIMARY KEY (rowid=?)}
 }
+
+do_execsql_test 4.2.0 {
+  DROP TABLE c1;
+  DROP TABLE c2;
+  DROP TABLE c3;
+  CREATE TABLE c1(k UNIQUE, v1);
+  CREATE TABLE c2(k UNIQUE, v2);
+  CREATE TABLE c3(k UNIQUE, v3);
+
+  INSERT INTO c1 VALUES(1, 2);
+  INSERT INTO c2 VALUES(2, 3);
+  INSERT INTO c3 VALUES(3, 'v3');
+
+  INSERT INTO c1 VALUES(111, 1112);
+  INSERT INTO c2 VALUES(112, 1113);
+  INSERT INTO c3 VALUES(113, 'v1113');
+}
+do_execsql_test 4.2.1 {
+  SELECT v1, v3 FROM c1 LEFT JOIN c2 ON (c2.k=v1) LEFT JOIN c3 ON (c3.k=v2);
+} {2 v3 1112 {}}
+do_execsql_test 4.2.2 {
+  SELECT v1, v3 FROM c1 LEFT JOIN c2 ON (c2.k=v1) LEFT JOIN c3 ON (c3.k=v1+1);
+} {2 v3 1112 {}}
+
+do_execsql_test 4.2.3 {
+  SELECT DISTINCT v1, v3 FROM c1 LEFT JOIN c2 LEFT JOIN c3 ON (c3.k=v1+1);
+} {2 v3 1112 {}}
+
+do_execsql_test 4.2.4 {
+  SELECT v1, v3 FROM c1 LEFT JOIN c2 LEFT JOIN c3 ON (c3.k=v1+1);
+} {2 v3 2 v3 1112 {} 1112 {}}
+
+do_eqp_test 4.2.5 {
+  SELECT v1, v3 FROM c1 LEFT JOIN c2 ON (c2.k=v1) LEFT JOIN c3 ON (c3.k=v2);
+} {
+  0 0 0 {SCAN TABLE c1} 
+  0 1 1 {SEARCH TABLE c2 USING INDEX sqlite_autoindex_c2_1 (k=?)}
+  0 2 2 {SEARCH TABLE c3 USING INDEX sqlite_autoindex_c3_1 (k=?)}
+}
+do_eqp_test 4.2.6 {
+  SELECT v1, v3 FROM c1 LEFT JOIN c2 ON (c2.k=v1) LEFT JOIN c3 ON (c3.k=v1+1);
+} {
+  0 0 0 {SCAN TABLE c1} 
+  0 1 2 {SEARCH TABLE c3 USING INDEX sqlite_autoindex_c3_1 (k=?)}
+}
 
 # 2017-11-23 (Thanksgiving day)
 # OSSFuzz found an assertion fault in the new LEFT JOIN eliminator code.
 #
 do_execsql_test 4.3.0 {
@@ -189,7 +234,35 @@
   SELECT a.x, c.x
     FROM t1 AS a
     LEFT JOIN t1 AS b ON (a.x=b.x)
     LEFT JOIN t2 AS c ON (a.x=c.x);
 } {1 {} 2 {} 3 {} 4 {} 5 {} 6 {} 7 {} 8 {} 9 {} 10 10}
+
+do_execsql_test 5.0 {
+  CREATE TABLE s1 (a INTEGER PRIMARY KEY);
+  CREATE TABLE s2 (a INTEGER PRIMARY KEY);
+  CREATE TABLE s3 (a INTEGER);
+  CREATE UNIQUE INDEX ndx on s3(a);
+}
+do_eqp_test 5.1 {
+  SELECT s1.a FROM s1 left join s2 using (a);
+} {
+  0 0 0 {SCAN TABLE s1}
+}
+do_eqp_test 5.2 {
+  SELECT s1.a FROM s1 left join s3 using (a);
+} {
+  0 0 0 {SCAN TABLE s1}
+}
+
+do_execsql_test 6.0 {
+  CREATE TABLE u1(a INTEGER PRIMARY KEY, b, c);
+  CREATE TABLE u2(a INTEGER PRIMARY KEY, b, c);
+  CREATE INDEX u1ab ON u1(b, c);
+}
+do_eqp_test 6.1 {
+  SELECT u2.* FROM u2 LEFT JOIN u1 ON( u1.a=u2.a AND u1.b=u2.b AND u1.c=u2.c );
+} {
+  0 0 0 {SCAN TABLE u2}
+}
 
 finish_test