Let
k
\mathbf {k}
be any field. Let
R
=
k
[
X
,
Y
,
Z
]
/
(
X
n
Y
−
Z
2
−
h
(
X
)
Z
)
R = \mathbf {k}[X,Y,Z]/(X^n Y - Z^2 - h(X)Z)
, where
h
(
0
)
≠
0
h(0) \ne 0
and
n
≥
2
n \geq 2
. We develop techniques for computing the AK invariant of a domain with arbitrary characteristic. We use these techniques to compute
AK
(
R
)
\operatorname {AK}(R)
, describe the automorphism group of
R
R
, and describe the isomorphism classes of these algebras. We then show that these algebras provide counterexamples to the cancellation problem over any field, extending Danielewski’s original counterexample.