Klingenberg planes are generalizations of Hjelmslev planes. If R is a local ring, one can construct a projective Klingenberg plane
V
(
R
)
{\textbf {V}}(R)
and a derived affine Klingenberg plane
A
(
R
)
{\textbf {A}}(R)
from R. If V is a projective Klingenberg plane, if
R
1
,
R
2
{R_1},\,{R_2}
and
R
3
{R_3}
are local rings, if
s
1
,
s
2
{s_1},\,{s_2}
and
s
3
{s_3}
are the sides of a nondegenerate triangle in V, and if each of the derived affine Klingenberg planes
a
(
V
,
s
i
)
\mathcal {a}\left ( {V,\,{s_i}} \right )
is isomorphic to
A
(
R
i
)
,
{\textbf {A}}({R_i}),\,
,
i
=
1
,
2
,
3
i\, = \,1,\,2,\,3
, then the rings
R
1
,
R
2
{R_1},\,{R_2}
and
R
3
{R_3}
are isomorphic, and V is isomorphic to
V
(
R
1
)
;
{\textbf {V}}({R_1});
; also, if g is a line of V, then the derived affine Klingenberg plane
a
(
V
,
g
)
\mathcal {a}({V,\,g})
is isomorphic to
A
(
R
1
)
\textbf {A}({R_1})
. Examples are given of projective Klingenberg planes V, each of which has the following two properties: (1) V is not isomorphic to
V
(
R
)
{\textbf {V}}(R)
for any local ring R; and (2) there is a flag
(
B
,
b
)
(B,\,b)
of V, and a local ring S such that each derived affine Klingenberg plane
a
(
V
,
m
)
\mathcal {a}({V,\,m})
is isomorphic to
A
(
S
)
{\textbf {A}}(S)
whenever
m
=
b
m\, = \,b
, or m is a line through B which is not neighbor to b.