We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A
0 and of the Weyl algebra A
1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras
A
h
=
〈
x
,
y
|
y
x
−
x
y
=
h
(
x
)
〉
,
{A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle ,
, where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah
. We also compute the Makar-Limanov invariant of absolute constants of Ah
over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah
.