  In this manuscript, we highlight a new phenomenon of complex algebraic
singularity formation for solutions of a large class of genuinely nonlinear
partial differential equations (PDEs). We start from a unique Cauchy datum,
which is holomorphic ramified around the smooth locus and is sufficiently
singular. Then, we expect the existence of a solution which should be
holomorphic ramified around the singular locus S defined by the vanishing of
the discriminant of an algebraic equation. Notice, moreover, that the monodromy
of the Cauchy datum is Abelian, whereas one of the solutions is non-Abelian.
Moreover, the singular locus S depends on the Cauchy datum in contrast to the
Leray principle (stated for linear problems only). This phenomenon is due to
the fact that the PDE is genuinely nonlinear and that the Cauchy datum is
sufficiently singular. First, we investigate the case of the inviscid Burgers
equation. Later, we state a general conjecture that describes the expected
phenomenon. We view this Conjecture as a working programme allowing us to
develop interesting new Mathematics. We also state another Conjecture 2, which
is a particular case of the general Conjecture but keeps all the flavour and
difficulty of the subject. Then, we propose a new algorithm with a map F such
that a fixed point of F would give a solution to the problem associated with
Conjecture 2. Then, we perform convincing, elaborate numerical tests that
suggest that a Banach norm should exist for which the mapping F should be a
contraction so that the solution (with the above specific algebraic structure)
should be unique. This work is a continuation of Leichtnam (1993).
