In the first part of our paper we discuss linear 2nd order differential equations in complex domain, especially Heun class equations, that is, equation and its confluent cases. The second is devoted to Painlev\'e I-VI equations. Our philosophy treat these families a unified way. This works well for We classification into 5 supertypes, subdivided 10 types (not counting trivial cases). also intro...

We propose a new algorithm for solving a system of two nonlinear transcendental equations with two complex variables based on the Müller algorithm. The two-dimensional Müller algorithm is tested on systems of different type and is found to work comparably to Newton’s method and Broyden’s method in many cases. The new algorithm is particularly useful in systems featuring the Heun functions whose...

The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d 2 /dx 2 + V (g; x), where the potential is an elliptic function depending on a coupling vector g ∈ R 4. Alternatively, this operator arises from the BC 1 specialization of the BC N elliptic nonrelativistic Calogero–Moser system (a.k.a. the Inozemtsev system). Under suitable restrict...

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. We also study the Poincar...