A generalization of the rigorous method of regularization is implemented to calculate the complex eigenvalues for a two dimensional arbitrarily shaped acoustically soft cavity with a longitudinal slit. The problem is reduced to the finding of non-trivial solutions of the coupled homogeneous well-conditioned Fredholm second kind infinite systems of linear algebraic equations that are solved numerically by the truncation method. The guaranteed fast convergence of the solution of the truncated system to the exact solution allows one to achieve any pre-determined accuracy by proper choice of truncation number. Formally, the complex eigenvalues coincide with the complex roots of the characteristic equation of the truncated infinite system. All calculations are performed with an accuracy of six significant decimal digits. The algorithm is free from limitations on the slit width, frequency band, and slit location along the bounding contour of a cavity. As an example, the spectrum of the complex eigenvalues for open elliptic cavity with moveable longitudinal slit is accurately investigated for various ellipse eccentricities, including the case of degenerated elliptic cavity - circular cavity. The slit width varies from zero value (closed cavity) to open semi-elliptic cavity.