We determine solutions of the Euler equation representing isolated vortices (monopoles, dipoles) in an infinite domain, for arbitrary values of energy, circulation, angular momentum and impulse. A linear relationship between vorticity and stream function is assumed inside the vortex (while the flow is irrotational outside). The emergence of these solutions in a turbulent flow is justified by the statistical mechanics of continuous vorticity fields. The additional restriction of mixing to a ‘maximum-entropy bubble’, due to kinetic constraints, is assumed. The linear relationship between vorticity and stream function is obtained from the statistical theory in the limit of strong mixing (when constraints are weak). In this limit, maximizing entropy becomes equivalent to a kind of enstrophy minimization. New stability criteria are investigated and imply in particular that, in most cases, the vorticity must be continuous (or slightly discontinuous) at the vortex boundary. Then, the vortex radius is automatically determined by the integral constraints and we can obtain a classification of isolated vortices such as monopoles and dipoles (rotating or translating) in terms of a single control parameter. This article generalizes the classification obtained in a bounded domain by Chavanis & Sommeria (1996).