  It is shown that the original Andrews--Curtis conjecture on balanced
presentations of the trivial group is equivalent to its "cyclic" version in
which, in place of arbitrary conjugations, one can use only cyclic
permutations. This, in particular, proves a satellite conjecture of Andrews and
Curtis made in 1966. We also consider a more restrictive "cancellative" version
of the cyclic Andrews--Curtis conjecture with and without stabilizations and
show that the restriction does not change the Andrews--Curtis conjecture when
stabilizations are allowed. On the other hand, the restriction makes the
conjecture false when stabilizations are not allowed.
