## 1 Introduction  
Optical frequency combs (OFCs) can be generated using mode-locked lasers, electro-optical modulation of continuous-wave light, or integrated microresonators [1]. When these combs are produced in an integrated microresonator, they are called Kerr combs. Kerr combs have recently attracted a large amount of attention in the photonics research community; they have applications in areas such as communications, metrology, optical clocks, photonic radars, artificial intelligence, and quantum information [2, 3, 4, 5, 6, 7]. Ensuring their reliability, robustness, and low cost is a key challenge. Dissipative Kerr solitons (DKSs) are a special type of Kerr comb. DKS yield a series of discrete, equally spaced frequency lines in the output spectrum with a constant frequency offset. DKS generation can be performed using a single microring resonator (MRR) [1, 2, 3, 8, 9, 10], defining paths in which the laser pump power and detuning are controlled [8, 11]. To understand the dynamics of this structure, the different operating regions of the MRR in the parameter space (\(\Delta_{r}\left|S\right|^{2}\)) have been identified [8, 12, 13]. These regions correspond to cnoidal waves, including Turing rolls, perfect soliton crystals, and single solitons or single DKSs, as well as chaotic solitons, soliton molecules, and imperfect soliton crystals. Specifically, [PERSON] et al. defined a deterministic path for single DKS generation in a Si\({}_{3}\)N\({}_{4}\) MRR, based on the above-mentioned regional classification [8]. However, it is difficult to apply this technique in experiments because it requires the pump laser power and wavelength to be changed rapidly along a defined path. Other approaches include thermal control, pulse-triggers, multiple laser pumps, and engineered spatial mode interactions, among others [1, 2, 14, 15]. Although these approaches can generate solitons deterministrically, they involve difficult and expensive experimental setups. In [16, 17], it was demonstrated that mode-crossings due to spatial mode-interactions can allow deterministic single soliton generation in a single MRR. However, its approach is conditioned by the structure's geometry, limiting achievable dispersion profile and constraining generation to the paths proposed by the study, which may not be optimal. One of the limitations of a single MRR is the inability to control mode interactions. Recently, the use of dual-coupled microring resonators (DCM) has been proposed to facilitate single DKS generation [18; 19; 20; 21]. This structure allows the formation of specific dispersion profiles that depend on the coupling of the auxiliary ring, thus facilitating the manufacturing and engineering of dispersion. DCM can be controlled to modify mode interactions and enlarge the region of the parameter space within which single solitons are stable, making them easier to obtain deterministrically. Approaches such as [18; 19; 20; 22; 23; 24; 25] demonstrate mathematical models and the dynamics governing light propagation in this structure. While these approaches demonstrate the formation of a DKS, those structures do not control mode interaction. As a result, their impact is fixed and confined to the specific case studies. Other work studies the stability and accessibility of DKSs in dual-coupled microresonators (DCMs) as the system parameters vary [26; 27; 28; 29]. These approaches have led to mathematical frameworks that facilitate the study of frequency combs, in particular enabling the determination of parameter regimes (within which stationary waveforms such as Turing rolls and DKSs exist) and their stability. However, a drawback in these studies is that the models are exclusively designed for microresonators with normal dispersion, overlooking structures with anomalous dispersion. Additionally, the structures analyzed do not control the mode interaction, restricting the generalization of the obtained results. In [22], the authors propose a DCM to generate Turing rolls and single DKS.