Let x be an indeterminate over ℂ. We investigate solutions \begin{eqnarray} \advance \displaywidth by -6pc \alpha(s,x)=\sum_{n\geq 0} \alpha_n(s)x^n,\nonumber \end{eqnarray}α(s,x)=∑n≥0αn(s)xn,α n : ℂ → ℂ, n ≥ 0, of the first cocycle equation \begin{eqnarray} \advance \displaywidth by -6pc \alpha (s+t,x)= \alpha (s,x)\alpha \bigl(t,F (s,x)\bigr),\qquad s,t\in\Complex, \hspace*{5cm}{\rm(Co1)}\nonumber \end{eqnarray}α(s+t,x)=α(s,x)α(t,F(s,x)), s,t∈C,(Co1)in ℂ [[x ]], the ring of formal power series over ℂ, where (F (s,x ))s ∈ ℂ is an iteration group of type II, i.e. it is a solution of the translation equation \begin{eqnarray} \advance \displaywidth by -6pc F(s+t,x)=F(s,F(t,x)),\qquad s,t\in\Complex, \hspace*{5cm}\rm(T)\nonumber \end{eqnarray} F ( s + t,x ) = F ( s,F ( t,x ) ) , s,t ∈ C , ( T ) of the form F (s,x ) ≡ x + c k (s )x k mod x k +1 , where k ≥ 2 and c k ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions α n (s ) of \begin{eqnarray} \advance \displaywidth by -6pc \alpha(s,x)=1+\sum_{n\geq 1}\alpha_n(s)x^n\nonumber \end{eqnarray}α(s,x)=1+∑n≥1αn(s)xnare polynomials in c k (s ).It is possible to replace this additive function c k by an indeterminate. Finally, we obtain a formal version of the first cocycle equation in the ring (ℂ [y ]) [[x ]] . We solve this equation in a completely algebraic way, by deriving formal differential equations or an Aczel–Jabotinsky type equation. This way it is possible to get the structure of the coefficients in great detail which are now polynomials. We prove the universal character of these polynomials depending on certain parameters, the coefficients of the generator K of a formal cocycle for iteration groups of type II. Rewriting the solutions Γ(y,x ) of the formal first cocycle equation in the form ∑n ≥ 1 ψ n (x )y n as elements of (ℂ [[x ]]) [[y ]], we obtain explicit formulas for ψ n in terms of the derivatives H (j ) (x ) and K (j ) (x ) of the generators H and K and also a representation of Γ(y,x ) similar to a Lie–Grobner series. There are interesting similarities between the solutions G (y,x ) of the formal translation equation for iteration groups of type II and the solutions Γ(y,x ) of the formal first cocycle equation for iteration groups of type II.