  It is shown that the nonlinear Ermakov-Milne-Pinney equation
$\rho^{\prime\prime}+v(x)\rho=a/\rho^3$ obeys the property of covariance under
a class of transformations of its coefficient function. This property is
derived by using supersymmetric, or Darboux, transformations. The general
solution of the transformed equation is expressed in terms of the solution of
the original one. Both iterations of these transformations and irreducible
transformations of second order in derivatives are considered to obtain the
chain of mutually related Ermakov-Milne-Pinney equations. The behaviour of the
Lewis invariant and the quantum number function for bound states is
investigated. This construction is illustrated by the simple example of an
infinite square well.
