In a 1995 paper N. Hitchin, using twistor methods, has constructed new solutions of the Einstein equations. They were based on the results of K. Tod, who had (a year earlier) shown that the anti-selfdual Einstein $SU(2)$-invariant metrics on the $4$-manifolds generically are reduced to the Painleve-$6$ equation, satisfied by the independent variable, parametrizing the $SU(2)$-orbits.  Hitchin expressed his solutions by rather cumbersome formulas it terms of theta-functions; their relation with the geometry of families of elliptic curves was not clarified in that paper.

However, in the case of algebraic solutions Hitchin has discovered this relation in a 2004 paper; it turned out to be expressible in terms of the Poncelet closure theorem. In the talk the corresponding isomonodromic families will be presented in several cases, including the simplest one where the twistor  space is the projectivisation  of the space of cubic polynomials.

A collection of these (and some others) ideas and constructions indicates the existence of a certain class of Einstein manifolds that have an arithmetic nature and are described completely by dessins d’enfants (the graphs, embedded into the oriented surfaces and related to arithmetic geometry by A. Grothendieck). Some related physical fantasies will possibly be mentioned.