I study the solutions of a particular family of Painlevé 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.

1 aMazzocco, Marta uhttp://hdl.handle.net/1963/311801068nas a2200121 4500008004300000245007400043210007000117260001300187520067000200100002000870700002000890856003600910 2000 en_Ud 00aMonodromy of certain Painlevé-VI transcendents and reflection groups0 aMonodromy of certain PainlevéVI transcendents and reflection gro bSpringer3 aWe 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.1 aDubrovin, Boris1 aMazzocco, Marta uhttp://hdl.handle.net/1963/288200388nas a2200109 4500008004100000245007500041210007000116260001000186653002600196100002000222856003600242 1998 en d00aAlgebraic Solutions to the Painlevé-VI Equation and Reflection Groups0 aAlgebraic Solutions to the PainlevéVI Equation and Reflection Gr bSISSA10aPainlevé VI equation1 aMazzocco, Marta uhttp://hdl.handle.net/1963/557401558nas a2200109 4500008004100000245007100041210006900112260001300181520119800194100002001392856003601412 1997 en d00aKam theorem for generic analytic perturbations of the Guler system0 aKam theorem for generic analytic perturbations of the Guler syst bSpringer3 aWe 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.1 aMazzocco, Marta uhttp://hdl.handle.net/1963/1038