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An entire solution of the Allen-Cahn equation $\Delta u=F'(u)$, where $F$ isan even, bistable function, is called a $2k$-end solution if its nodal set isasymptotic to $2k$ half lines, and if along each of these half lines thefunction $u$ looks like the one dimensional, heteroclinic solution. In thispaper we initiate a program to classify the four-end solutions of theAllen-Cahn equation in $\R^2$. We show that there exists a one parameter familyof solutions containing the saddle solution, for which the angle between thenodal lines is $\frac{\pi}{2}$, as well as solutions for which the anglebetween the asymptotic half lines is any $\theta\in (0, \frac{\pi}{2})$. Thisjustifies the definition of the angle map for a four-end solution $u$, which isthe angle $\theta=\theta(u)\in (0, \frac{\pi}{2})$ between the asymptote to thenodal line in the first quadrant and the x axis. Then we show that on anyconnected component in the moduli space of four-end solutions the angle map issurjective onto $(0,\frac{\pi}{2})$.

The space of 4-ended solutions to the Allen-Cahn equation on the plane