  This paper is devoted to the classification of the infinite families of
Teichmuller curves generated by Prym eigenforms of genus 3 having a single
zero. These curves were discovered by McMullen. The main invariants of our
classification is the discriminant D of the corresponding quadratic order, and
the generators of this order. It turns out that for D sufficiently large, there
are two Teichmueller curves when D=1 modulo 8, only one Teichmueller curve when
D=0,4 modulo 8, and no Teichmueller curves when D=5 modulo 8. For small values
of D, where this classification is not necessarily true, the number of
Teichmueller curves can be determined directly. The ingredients of our proof
are first, a description of these curves in terms of prototypes and models, and
then a careful analysis of the combinatorial connectedness in the spirit of
McMullen. As a consequence, we obtain a description of cusps of Teichmueller
curves given by Prym eigenforms.
  We would like also to emphasis that even though we have the same statement
compared to, when D=1 modulo 8, the reason for this disconnectedness is
different.
  The classification of these Teichmueller curves plays a key role in our
investigation of the dynamics of SL(2,R) on the intersection of the Prym
eigenform locus with the stratum H(2,2), which is the object of a forthcoming
paper.
