This paper explores some nonlinear systems of singular partial differential equations written in the form t A D t U = Λ(t, x)U + f t, x, ζU, t A D x U. Under an assumption on Λ, unique solvability theorems are provided in the space of functions that are holomorphic in x on an open set, differentiable with respect to t on a real interval ]0, r] and extending to a continuous function at t = 0. The studied systems contain Fuchsian systems.