The aim of the present article is to construct quadratically integrable three dimensional systems in non-vanishing magnetic fields which possess so-called non-subgroup type integrals. The presence of such integrals means that the system possesses a pair of integrals of motion in involution which are (at most) quadratic in momenta and whose leading order terms, that are necessarily elements of the enveloping algebra of the Euclidean algebra, are not quadratic Casimir operators of a chain of its subalgebras. By imposing in addition that one of the integrals has the leading order term L 2 z we can consider three such commuting pairs: circular parabolic, oblate spheroidal and prolate spheroidal. We find all possible integrable systems possessing such structure of commuting integrals and describe their Hamiltonians and their integrals.We show that our assumptions imply the existence of a first order integral L z , i.e. rotational invariance, of all such systems. As a consequence, the Hamilton-Jacobi equation of each of these systems with magnetic field separates in the corresponding coordinate system, as it is known to be the case for all quadratically integrable systems without magnetic field, and in contrast with the subgroup type, i.e. Cartesian, spherical and cylindrical, cases, with magnetic fields.We also look for superintegrable systems within the circular parabolic integrable class. Assuming the additional integral to be first order we demonstrate that only previously known systems exist. However, for a particular second order ansatz for the sought integral (L 2 + . . .) we find a minimally quadratically superintegrable system. It is not quadratically maximally superintegrable but appears to possess bounded closed trajectories, hinting at hypothetical higher order superintegrability.

Abstract. The objective of this paper is to formulate two distinct supersymmetric (SUSY) extensions of the Gauss-Weingarten and Gauss-Codazzi (GC) equations for conformally parametrized surfaces immersed in a Grassmann superspace, one in terms of a bosonic superfield and the other in terms of a fermionic superfield. We perform this analysis using a superspace-superfield formalism together with a SUSY version of a moving frame on a surface. In constrast with the classical case, where we have three GC equations, we obtain six such equations in the bosonic SUSY case and four such equations in the fermionic SUSY case. In the fermionic case the GC equations resemble the form of the classical GC equations. We determine the Lie symmetry algebra of the classical GC equations to be infinite-dimensional and perform a subalgebra classification of the one-dimensional subalgebras of its largest finite-dimensional subalgebra. We then compute superalgebras of Lie point symmetries of the bosonic and fermionic SUSY GC equations respectively, and classify the onedimensional subalgebras of each superalgebra into conjugacy classes. We then use the symmetry reduction method to find invariants, orbits and reduced systems for two one-dimensional subalgebras for the classical case, two one-dimensional subalgebras for the bosonic SUSY case and two one-dimensional subalgebras for the fermionic SUSY case. We find explicit solutions of these reduced SUSY systems, which correspond to different surfaces immersed in a Grassmann superspace. Within this framework for the SUSY versions of the GC equations, a geometrical interpretation of the results is discussed.

We extend the investigation of three-dimensional Hamiltonian systems of non-subgroup type admitting non-zero magnetic fields and an axial symmetry, namely the circular parabolic case, the oblate spheroidal case and the prolate spheroidal case. More precisely, we focus on linear and some special cases of quadratic superintegrability. In the linear case, no new superintegrable system arises. In the quadratic case, we found one new minimally superintegrable system that lies at the intersection of the circular parabolic and cylindrical cases and another one at the intersection of the cylindrical, spherical, oblate spheroidal and prolate spheroidal cases. By imposing additional conditions on these systems, we found for each quadratically minimally superintegrable system a new infinite family of higher-order maximally superintegrable systems. These two systems are linked respectively with the caged and harmonic oscillators without magnetic fields through a time-dependent canonical transformation.

Abstract. The paper presents the bosonic and fermionic supersymmetric extensions of the structural equations describing conformally parametrized surfaces immersed in a Grasmann superspace, based on the authors' earlier results. A detailed analysis of the symmetry properties of both the classical and supersymmetric versions of the Gauss-Weingarten equations is performed. A supersymmetric generalization of the conjecture establishing the necessary conditions for a system to be integrable in the sense of soliton theory is formulated and illustrated by the examples of supersymmetric versions of the sine-Gordon equation and the Gauss-Codazzi equations.

Abstract. In this paper, we construct and investigate two supersymmetric versions of the Fokas-Gel'fand formula for the immersion of 2D surfaces associated with a supersymmetric integrable system. The first version involves an infinitesimal deformation of the zero-curvature condition and the linear spectral problem associated with this system. This deformation leads the surfaces to be represented in terms of a bosonic supermatrix immersed in a Lie superalgebra. The second supersymmetric version is obtained by using a fermionic parameter deformation to construct surfaces expressed in terms of a fermionic supermatrix immersed in a Lie superalgebra. For both extensions, we provide a geometrical characterization of deformed surfaces using the super Killing form as an inner product and a super moving frame formalism. The theoretical results are applied to the supersymmetric sine-Gordon equation in order to construct super soliton surfaces associated with five different symmetries. We find integrated forms of these surfaces which represent constant Gaussian curvature surfaces and nonlinear Weingarten-type surfaces.