, Let b be a branch satisfying condition (ii-a) of Theorem 4.1. Then its base point C is a regular point of the conic I, and b is tangent to I. We treat the two following cases separately. Case 1): I is a union of two lines. Then b has local relative projective symmetry property of type A-w, by Proposition 4.16, Subcase 3a). Hence, it is quadratic, by Theorem 4.17. Case 2): I is a regular conic, Hence, it is quadratic, vol.17

, Rationally integrable I-angular billiards. Proof of Theorem 1, p.25

I. Let, CP 2 be a conic (regular or a pair of distinct lines), and let ? ? CP 2 be an irreducible algebraic curve different from a line and from I and generating a rationally integrable I-angular billiard

, All the singular and inflection points (if any) of the curve ? lie in I

, Namely, for every C 2 -smooth arc ? ? ?? with non-zero geodesic curvature the statement of Theorem 6.1 is proved there for each non-linear irreducible component ? of the Zariski closure of the ?-dual curve ? * . But the proofs given in [10, 11] remain valid in the general context of Theorem 6.1. Each local branch of the curve ? at a base point in ? ? I that satisfies the conditions of some of the statements (i), (ii-a), or (ii-b) of Theorem 4.1 also satisfies the corresponding statement

?. Let, Let ?(M ) be its non-trivial homogeneous polynomial integral of even degree 2n: M = [r, v], and ?([r, v]) is not a function of the squared norm ||v|| 2 =< Av, v > in the metric of the surface ?, One has ?(M ) ? c < AM, M > n , since < AM, M >=< Av

A. M. Abdrakhmanov, Integrable billiards, Mosc. Univ. Mech. Bull, vol.45, issue.6, pp.13-17, 1990.

A. M. Abdrakhmanov, On integrable systems with elastic reflections (in Russian), Mosc. Univ. Mech. Bull, vol.45, issue.5, pp.14-16, 1990.

L. Advis-gaete, B. Carry, M. Gualtieri, C. Guthmann, E. Reffet et al., Golfer's dilemma, Am. J. Phys, vol.74, issue.6, pp.497-501, 2006.

A. Avila, J. De-simoi, and V. Kaloshin, An integrable deformation of an ellipse of small eccentricity is an ellipse, Ann. of Math, issue.2, pp.527-558, 2016.

E. Amiran, Caustics and evolutes for convex planar domains, J. Diff. Geometry, vol.28, pp.345-357, 1988.

M. Berger, Seules les quadriques admettent des caustiques, Bull. Soc. Math. France, vol.123, pp.107-116, 1995.

M. Bialy, Convex billiards and a theorem by E, Hopf. Math. Z, vol.214, issue.1, pp.147-154, 1993.

M. Bialy, Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane, Discrete Contin. Dyn. Syst, vol.33, issue.9, pp.3903-3913, 2013.

M. Bialy, On totally integrable magnetic billiards on constant curvature surface, Electron. Res. Announc. Math. Sci, vol.19, pp.112-119, 2012.

M. Bialy and A. E. Mironov, Angular billiard and algebraic Birkhoff conjecture, Adv. in Math, vol.313, pp.102-126, 2017.

M. Bialy and A. E. Mironov, Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane, J. Geom. Phys, vol.115, pp.150-156, 2017.

M. Bialy and A. E. Mironov, On fourth-degree polynomial integrals of the Birkhoff billiard, Proc. Steklov Inst. Math, vol.295, issue.1, pp.27-32, 2016.

M. Bialy and A. E. Mironov, Algebraic non-integrability of magnetic billiards, J. Phys. A, vol.49, issue.45, p.pp, 2016.

M. Bialy and A. E. Mironov, A survey on polynomial in momenta integrals for billiard problems, Issue 2131, vol.336, 2018.

S. V. Bolotin, First integrals of systems with gyroscopic forces. (Russian) Vestnik Moskov, Univ. Ser. I Mat. Mekh, vol.113, issue.6, pp.75-82, 1984.

S. V. Bolotin, Integrable Birkhoff billiards. Mosc. Univ. Mech. Bull, vol.45, issue.2, pp.10-13, 1990.

S. V. Bolotin, Integrable billiards on surfaces of constant curvature, Math. Notes, vol.51, issue.1-2, pp.117-123, 1992.

E. Brieskorn and H. Knörrer, Plane algebraic curves, 1986.

A. Delshams and R. Ramirez-ros, On Birkoff 's [Birkhoff 's] conjecture about convex billiards, Proceedings of the 2nd Catalan Days on Applied Mathematics, pp.85-94, 1995.

V. Dragovi? and M. Radnovi?, Integrable billiards and quadrics, Russian Math. Surveys, vol.65, issue.2, pp.319-379, 2010.

V. Dragovi? and M. Radnovi?, Bicentennial of the great Poncelet theorem (1813-2013): current advances, Bull. Amer. Math. Soc. (N.S.), vol.51, issue.3, pp.373-445, 2014.

V. Dragovi? and M. Radnovi?, Pseudo-integrable billiards and arithmetic dynamics, J. Mod. Dyn, vol.8, issue.1, pp.109-132, 2014.

V. Dragovi? and M. Radnovi?, Periods of pseudo-integrable billiards, Arnold Math. J, vol.1, issue.1, pp.69-73, 2015.

V. Dragovi? and M. Radnovi?, Pseudo-integrable billiards and double reflection nets, Russian Math. Surveys, vol.70, issue.1, pp.1-31, 2015.

A. Glutsyuk, On quadrilateral orbits in complex algebraic planar billiards, Moscow Math. J, vol.14, issue.2, pp.239-289, 2014.

A. A. Glutsyuk, On two-dimensional polynomially integrable billiards on surfaces of constant curvature, Doklady Mathematics, vol.98, issue.1, pp.382-385, 2018.
URL : https://hal.archives-ouvertes.fr/ensl-01964938

A. Glutsyuk and E. Shustin, On polynomially integrable planar outer billiards and curves with symmetry property, Math. Annalen, vol.372, pp.1481-1501, 2018.
URL : https://hal.archives-ouvertes.fr/ensl-01413589

G. Greuel, C. Lossen, and E. Shustin, Introduction to singularities and deformations, 2007.

P. Griffiths and J. Harris, Principles of algebraic geometry, vol.1, 1978.

H. Hironaka, Arithmetic genera and effective genera of algebraic curves, Mem. Coll. Sci. Univ. Kyoto, Sect, vol.30, pp.177-195, 1956.

V. Kaloshin and A. Sorrentino, On local Birkhoff Conjecture for convex billiards, Ann. of Math, vol.188, issue.1, pp.315-380, 2018.

V. V. Kozlov and N. V. Denisova, Symmetries and topology of dynamical systems with two degrees of freedom, Russian Acad. Sci. Sb. Math, vol.80, issue.1, pp.105-124, 1995.

V. V. Kozlov, D. V. Treshchev, and . Billiards, A genetic introduction to the dynamics of systems with impacts, Translated from Russian by J.R.Schulenberger. Translations of Mathematical Monographs, vol.89, 1991.

V. F. Lazutkin, The existence of caustics for a billiard problem in a convex domain, Math. USSR Izvestija, vol.7, pp.185-214, 1973.

J. Marco, Entropy of billiard maps and a dynamical version of the Birkhoff conjecture, J. Geom. Phys, vol.124, pp.413-420, 2018.

J. Milnor, Singular points of complex hypersurfaces, 1968.

H. Poritsky, The billiard ball problem on a table with a convex boundary -an illustrative dynamical problem, Ann. of Math, issue.2, pp.446-470, 1950.

A. Ramani, A. Kalliterakis, B. Grammaticos, and B. Dorizzi, Integrable curvilinear billiards, Phys. Lett. A, vol.115, issue.1-2, pp.25-28, 1986.

E. Shustin, On invariants of singular points of algebraic curves, Math. Notes, vol.34, issue.5-6, pp.962-963, 1983.

S. Tabachnikov, Geometry and billiards. Student Mathematical Library, Mathematics Advanced Study Semesters, vol.30, 2005.

S. Tabachnikov, On algebraically integrable outer billiards, Pacific J. of Math, vol.235, issue.1, pp.101-104, 2008.

D. Treschev, Billiard map and rigid rotation, Phys. D, pp.31-34, 2013.

D. Treschev, On a Conjugacy Problem in Billiard Dynamics, Proc. Steklov Inst. Math, vol.289, issue.1, pp.291-299, 2015.

D. Treschev, A locally integrable multi-dimensional billiard system, Discrete Contin. Dyn. Syst, vol.37, issue.10, pp.5271-5284, 2017.

A. P. Veselov, Integrable systems with discrete time, and difference operators, Funct. Anal. Appl, vol.22, issue.2, pp.83-93, 1988.

A. P. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys, vol.7, pp.81-107, 1990.

M. P. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom, vol.40, issue.1, pp.155-164, 1994.