  We are interested at first in the study of the monotonicity for the period
function of the conservative equation \ $(1)\quad \ddot x + g(x) = 0.$\quad
Some refinements of known criteria are brought. Moreover, we give necessary and
sufficient conditions so that the analytic potential of equation $(1)$ is
isochronous. These conditions which are different from those introduced firstly
by Koukles and Piskounov and thereafter by Urabe appear sometime to be easier
to use. We then apply these results to produce families of isochronous
potentials depending on many parameters, some of them are news. Moreover,
analytic isochronicity requirements of parametrized potentials will also be
considered
