  One-variable thermostats are studied as a generalization of the
Nos\'e--Hoover method which is aimed to achieve Gibbs' canonical distribution
with conserving the time-reversibility. A condition for equations of motion for
the system with the thermostats is derived in the form of a partial
differential equation. Solutions of this equation construct a family of
thermostats including the Nos\'e--Hoover method as the minimal solution. It is
shown that the one-variable thermostat coupled with the one-dimensional
harmonic oscillator loses its ergodicity with large enough relaxation time. The
present result suggests that multi-variable thermostats are required to assure
the ergodicity and to work as heatbath.
