Let \(\Phi\) be a local diffeomorphism on \(\mathbb{K}^n\) (where \(\mathbb{K}\) is \(\mathbb{R}\) or \(\mathbb{C}\)) having the origin as its isolated fixed point. This diffeomorphism is called integrable if there exists \(p\geq1\) pairwise commuting (germs of) diffeomorphisms \(\Phi_1=\Phi\), \(\Phi_2\), \(\dots\), \(\Phi_p\) with \(d\Phi_i(0)=A_i\) and \(q=n-p\) common first integrals \(F_1,\dots,F_q\) of those diffeomorphisms such that:
(i) The diffeomorphisms \(\Phi_i\) are independent, i.e., the matrices \(\{\ln A_i\}\), \(i=1,\dots,p\), are linearly independent over \(\mathbb{K}^n\) (for \(\mathbb{K}=\mathbb{C}\) independence is required for the families of all possible logarithms);
(ii) The first integrals \(F_j\) are functionally independent almost everywhere.
In this case we say that \((\Phi_1,\dots,\Phi_p,F_1,\dots,F_q)\) is a discrete completely integrable system of type \((p,q)\).
The authors consider the normal form problem of a commutative family of germs of analytic or smooth diffeomorphisms at a fixed point and give sufficient conditions which ensure that such an integrable family can be transformed into a normal form by an analytic or smooth transformation if the initial diffeomorphisms are respectively analytic or smooth.