  We develop a qualitative theory for real solutions of the equation $y''=6y^2
-x$. In this work a restriction $x\leq0$ is assumed. An important ingredient of
our theory is the introduction of several new transcendental functions of one,
two, and three variables that describe different properties of the solutions.
In particular, the results obtained allow us to completely analyse the
Dirichlet boundary value problem $y(a)=y^0$, $y(b)=y_0$ for $a<b\leq0$.
