We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space $ℂ{\mathbb{P}}^n$ for every dimension $n\ge 2$ and every degree $d\ge 2$. Precisely, we construct a foliation $ℱ$ which is induced by a homogeneous vector field of degree $d$, has a finite singular set and all the regular leaves are dense in the whole of $ℂ{\mathbb{P}}^n$. Moreover, $ℱ$ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if $ℱ$ is conjugate to another holomorphic foliation...

In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.