I study the solutions of a particular family of Painlev{\'e} VI equations with the parameters $\beta=\gamma=0, \delta=1/2$ and $2\alpha=(2\mu-1)^2$, for $2\mu\in\mathbb{Z}$. I show that the case of half-integer $\mu$ is integrable and that the solutions are of two types: the so-called Picard solutions and the so-called Chazy solutions. I give explicit formulae for them and completely determine their asymptotic behaviour near the singular points $0,1,\infty$ and their nonlinear monodromy. I study the structure of analytic continuation of the solutions to the PVI$\mu$ equation for any $\mu$ such that $2\mu\in\mathbb{Z}$. As an application, I classify all the algebraic solutions. For $\mu$ half-integer, I show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For $\mu$ integer, I show that all algebraic solutions belong to a one-parameter family of rational solutions.

}, doi = {10.1007/PL00004500}, url = {http://hdl.handle.net/1963/3118}, author = {Marta Mazzocco} } @article {2000, title = {Monodromy of certain Painlev{\'e}-VI transcendents and reflection groups}, journal = {Invent. Math. 141 (2000) 55-147}, number = {arXiv.org;math/9806056v1}, year = {2000}, publisher = {Springer}, abstract = {We study the global analytic properties of the solutions of a particular family of Painleve\\\' VI equations with the parameters $\\\\beta=\\\\gamma=0$, $\\\\delta={1\\\\over2}$ and $\\\\alpha$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painleve\\\' VI equation.}, doi = {10.1007/PL00005790}, url = {http://hdl.handle.net/1963/2882}, author = {Boris Dubrovin and Marta Mazzocco} } @mastersthesis {1998, title = {Algebraic Solutions to the Painlev{\'e}-VI Equation and Reflection Groups}, year = {1998}, school = {SISSA}, keywords = {Painlev{\'e} VI equation}, url = {http://hdl.handle.net/1963/5574}, author = {Marta Mazzocco} } @article {1997, title = {Kam theorem for generic analytic perturbations of the Guler system}, journal = {Z. Angew. Math. Phys. 48 (1997), no. 2, 193-219}, number = {SISSA;136/95/FM}, year = {1997}, publisher = {Springer}, abstract = {We apply here KAM theory to the fast rotations of a rigid body with a fixed point, subject to a purely positional potential. The problem is equivalent to a small perturbation of the Euler system. The difficulty is that the unperturbed system is properly degenerate, namely the unperturbed Hamiltonian depends only on two actions. Following the scheme used by Arnol\\\'d for the N-body problem, we use part of the perturbation to remove the degeneracy: precisely, we construct Birkhoff normal form up to a suitable finite order, thus eliminating the two fast angles; the resulting system is nearly integrable and (generically) no more degenerate, so KAM theorem applies. The resulting description of the motion is that, if the initial kinetic energy is sufficiently large, then for most initial data the angular momentum has nearly constant module, and moves slowly in the space, practically following the level curves of the initial potential averaged on the two fast angles; on the same time the body precesses around the instantaneous direction of the angular momentum, essentially as in the Euler-Poinsot motion. We also provide two simple physical examples, where the procedure does apply.}, doi = {10.1007/PL00001474}, url = {http://hdl.handle.net/1963/1038}, author = {Marta Mazzocco} }