New classes of quadratically integrable systems with velocity dependent potentials: non-subgroup type cases

Integrable systems in magnetic fields: the generalized parabolic cylindrical case

“…upon a suitable rotation of the coordinate system and a linear combination of integrals X 1 , X 2 and H. As we are mainly interested in the integrals of the generalized form, we will hereafter assume that at least one of the parameters a, b, c is nonvanishing. As we noted for the elliptic cylindrical case in [12], the determining equations for the standard integrals (i.e. a = b = c = 0) are less overdetermined and therefore harder to solve.…”

“…We call such integrals nonstandard. To confirm that these classes are not empty, we have to consider the lower order terms in momenta as well in each class; we have done it for some of them [11,12,16]. Quite a few nonstandard integrable systems were found, though a complete study of all classes has not been achieved yet.…”

“…In this paper, we consider one of the more complicated classes of [21], case (j), which generalizes the parabolic cylindrical structure of integrals by adding leading order terms that depend on three additional parameters. Our goal is to classify the integrable systems in this class as well as to shed more light on the conjecture in our previous papers [12,16]: In the presence of magnetic fields, there appears to be an asymmetry between the subgroup-and non-subgroup-type coordinates 3 already at the level of quadratic integrals.…”

Integrable systems of the ellipsoidal, paraboloidal and conical type with magnetic field

“…(β) 2 , (6.11) and one subcase that resists full classification but we were able to obtain some particular solutions. This situation is reminiscent of the elliptic and parabolic type integrable systems in two dimensions and the corresponding three dimensional elliptic and parabolic cylindrical cases for which full classification is also not known, see [13,29].…”

“…In [28] we constructed all integrable systems of one nonstandard case of [26] (case (i) therein), leading to one family of solutions depending on 5 parameters. In [29] we considered further four classes with generalized structure of the integrals, showing that for the prolate / oblate spheroidal, circular parabolic and nonstandard class (g) of [26] the inclusion of additional terms does not lead to any new system in addition to the standard structures of [17], and fully classified systems with generalized elliptic cylindrical integrals. Curiously, the standard elliptic cylindrical case (as well as parabolic cylindrical one) remain unclassified.…”