In this paper the conditions for the existence of one invariant straight line and one invariant cubic in a cubic differential system with a singular point of a center or a focus type, when these curves have not intersecting points, are found. It is proved that the singular point is a center if and only if the first five
Lyapunov quantities vanish. The center conditions were determined by using the method of Darboux integrability.

[1] Amel’kin V. V., Lukashevich N. A., Sadovskii A. P. Non-linear oscillations in the systems of second order. Belarusian University Press, Belarus, 1982.
[2] Chavarriga J., Gin´e J. Integrability of cubic systems with degenerate infinity. Differential Equations Dynamical Systems, 1988, 6, 425-–438.
[3] Cozma D. The problem of the center for cubic systems with two parallel invariant straight lines and one invariant conic. Nonlinear Differ. Equ. and Appl., 2009, 16, 213–234.
[4] Cozma D. The problem of the centre for cubic differential systems with two homogeneous invariant straight lines and one invariant conic. Annals of Differential Equations, 2010, 26 (4), 385–399.
[5] Cozma D. Center problem for a class of cubic systems with a bundle of two invariant straight lines and one invariant conic. Buletinul Academiei de Științe a Republicii Moldova. Matematica, 2010, 64 (3), 51–66.
[6] Cozma D. Integrability of cubic systems with invariant straight lines and invariant conics. Știința, Chișinău, 2013.
[7] Cozma D. Centers in a cubic differential system with one invariant straight line. ROMAI Journal, 2011, 7 (2), 53–62.
[8] Cozma D. The problem of the center for cubic systems with two parallel invariant straight lines and one invariant cubic. ROMAI Journal, 2015, 11 (2), 63–75.
[9] Cozma D., Dascalescu A. Center conditions for a cubic system with a bundle of two invariant straight lines and one invariant cubic. ROMAI Journal, 2017, 13 (2), 39–54.
[10] Cozma D., Dascalescu A. Integrability conditions for a class of cubic differential systems with a bundle of two invariant straight lines and one invariant cubic. Buletinul Academiei de Științe a Republicii Moldova. Matematica, 2018, 86 (1), 120–138.
[11] Cozma D., Matei A. Center conditions for a cubic differential system having an integrating factor. Bukovinian Math. Journal, 2020, 8 (2), 6–13. 
[12] Cozma D., Matei A. Integrating factors for a cubic differential system with two algebraic solutions. ROMAI Journal, 2021, 17 (1), 65–86.
[13] Cozma D. Darboux integrability of a cubic differential system with two parallel invariant straight lines. Carpathian J. Math., 2022, 38 (1), 129–137.
[14] Cozma D. First integrals in a cubic differential system with one invariant straight line and one invariant cubic. Acta et Commentationes, Exact and Natural Sciences, 2023, 16 (2), 97-–105.
[15] Dascalescu A. Center conditions for a cubic differential system with two invariant straight lines and one invariant cubic. Bukovinian Math. Journal, 2018, 6 (3-4), 53–62. DOI: https://doi.org/10.31861/bmj2018.03.053
[16] Dukari´c M. On integrability and cyclicity of cubic systems. Electr. J. Qual. Theory Differ. Equ., 2020, 55, 1–19.
[17] Llibre J. On the centers of cubic polynomial differential systems with four invariant straight lines. Topological Methods in Nonlinear Analysis, 2020, 55 (2), 387–402.
[18] Lyapunov A.M. The general problem of stability of motion. Gostekhizdat, Moscow, 1950.
[19] Romanovski V.G., Shafer D.S. The center and cyclicity problems: a computational algebra approach, Birkh¨auser, Boston, Basel, 2009.
[20] Sadovskii A.P., Shcheglova T.V. Solution of the center-focus problem for a nineparameter cubic system. Differential Equations, 2011, 47 (2), 208–223.
[21] Schlomiuk D. Algebraic and geometric aspects of the theory of polynomial vector fields, in Bifurcations and periodic orbits of vector fields (D. Schlomiuk, Ed.), Kluwer Academic Publishes, 1993, pp. 429-467.
[22] Șubă A., Cozma D. Solution of the problem of center for cubic differential systems with three invariant straight lines in generic position. Qualitative Theory of Dynamical Systems, 2005, 6, 45–58.
[23] Xiang Zhang. Integrability of Dynamical Systems: Algebra and Analysis. Singapure: Springer Nature Singapure, 2017.