  We study the problem of the existence and the holomorphicity of the
Monge-Amp\`ere foliation associated to a plurisubharmonic solutions of the
complex homogeneous Monge-Amp\`ere equation even at points of arbitrary
degeneracy. We obtain good results for real analytic unbounded solutions. As a
consequence we also provide a positive answer to a question of Burns on
homogeneous polynomials whose logarithm satisfies the complex Monge-Amp\`ere
equation and we obtain a generalization the work of P.M. Wong on the
classification of complete weighted circular domains.
