of conserved quantum numbers, certain channels will be coupled by the nucleus-nucleus interaction and we use labels $\kappa, \lambda, \mu, \nu$ for these channels. Thus the matrix element of the nucleus-nucleus interaction $V$ between the wavefunctions for channels $\kappa$ and $\lambda$, corresponding to integrating over all coordinates but $r$, will be written $V_{\kappa\lambda}(r)$. The increment $\Delta S_{\kappa\lambda}$ in the non-diagonal S-matrix which is due to a small perturbation $\Delta V_{\kappa\lambda}(r)$, is$$\Delta S_{\kappa\lambda} = \frac{{\rm i}\mu}{\hbar^2 k} 
 \sum_{\mu\nu}\int_0^{\infty}\psi_{\mu\kappa}(r) \Delta V_{\mu\nu}(r)
\psi_{\nu\lambda} {\rm d}r \label{integ}$$ where $\psi_{\nu\kappa}$ is the $\nu$th channel (first index) component of that coupled channel solution for the unperturbed non-diagonal potential for which there is in-going flux in channel $\kappa$ (second index) only. The normalisation is $\psi_{\nu\kappa} \rightarrow \delta_{\kappa\nu} I_{l_{\kappa}} - S_{\nu\kappa} 
O_{l_{\nu}}$ where $I_l$ and $O_l$ are incoming and outgoing Coulomb wavefunctions for orbital angular momentum $l$; there is no complex conjugation in the integral. Starting from Eq.\[integ\], spin-one inversion becomes a straightforward generalisation of the procedure outlined above and described in Refs. [@ketal1; @zuev; @candm]. The method is implemented in the code IMAGO [@imago] where the linearity relations have been exhaustively tested by the gradient method.

A convenient feature of the IP method is that one can 