  We classify 'primitive normal compactifications' of C^2 (i.e. normal analytic
surfaces containing C^2 for which the curve at infinity is irreducible),
compute the moduli space of these surfaces and their groups of auomorphisms. In
particular we show that in 'most' of these surfaces C^2 is 'rigidly embedded'.
As an application we give a description of 'embedded isomorphism classes' of
planar curves with one place at infinity. We also compute the canonical divisor
of these surfaces; it turns out that their log discrepancy is related to the
Frobenius number of the semigroup of poles along the curve at infinity. We use
the computation to classify Gorenstein primitive compactifications of C^2 with
rational and minimally elliptic singularities, extending a result of Brenton,
Drucker and Prins (Ann. of Math. Stud., vol 100, 1981). As another application
we characterize weighted projective spaces of the form P^2(1,1,q) in terms of
their 'log discrepancy' and 'index', generalizing a characterization of P^2 due
to Borisov (Journal of Algebraic Combinatorics, 2014).
