  We employ the modification of the basic Penrose formula in twistor theory,
which allows to introduce commuting composite space-time coordinates. It
appears that in the course of such modification the internal symmetry SU(2) of
two-twistor system is broken to U(1). We consider the symplectic form on
two-twistor space, permitting to interpret its 16 real components as a
phase-space. After a suitable change of variables such a two-twistor phase
space is split into three mutually commuting parts, describing respectively the
standard relativistic phase space (8 degrees of freedom), the spin sector (6
degrees of freedom) and the canonical pair angle-charge describing the electric
charge sector (2 degrees of freedom). We obtain a geometric framework providing
a twistor-inspired 18-dimensional extended relativistic phase space
$\mathcal{M}^{18}$. In such a space we propose the action only with first class
constraints, describing the relativistic particle characterized by mass, spin
and electric charge.
