§\[subsec:poly\_study\], in certain bidimensional cases (specifically, when the constant of motion $\alpha_1$ is positive) the polynomial $p_\xi\left(\xi\right)$ might have no real zeroes, and thus the simplified formula cannot be used. While in this case it is still possible to proceed to a complete solution via the full formula in conjunction with the general theory for the integration of rational functions of elliptic functions (see @halphen_traite_1886 [chapter VII] and @greenhill_applications_1959 [chapter VII]), the resulting expressions for $\phi\left(\tau\right)$ and $t\left(\tau\right)$ will be more complicated than the formulæ obtained for the three-dimensional case.

An additional complication in the bidimensional case is the presence of the discontinuity discussed in §\[subsec:poly\_study\]. In correspondence of a polar transit, either $p_\xi$ or $p_\eta$ will switch sign. This discontinuity must be taken into account in the computation and inversion of the integral , and ultimately it has the effect of introducing a branching in the solutions for $\xi\left(\tau\right)$ and $\eta\left(\tau\right)$.

Quasi-periodicity and periodicity
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Our solution to the Stark problem is based on the Weierstrass elliptic and related functions. Without giving a full account of the theory of the Weierstrassian functions (for which we refer to standard textbooks such as @whittaker_course_1927), we will recall here briefly a few 