  The problem we consider is to define families of n-dimensional integrals,
endowed with group actions as in Rhin-Viola's work on irrationality measures of
$\zeta(2)$ and $\zeta(3)$, the values of which are linear forms, over the
rationals, in multiple zeta values of weight at most n. We generalize
Vasilyev's and Sorokin's approaches, and give a change of variables that
connects them to each other. We describe a group structure for a n-dimensional
integral that specializes, for n=2 and n=3, to the ones obtained by Rhin and
Viola.
