We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in [A. González-López, N. Kamran and P.J. Olver, Proc. London Math. Soc. 64, 339 (1992)] and we interpret their results as a local classification of Lie systems. Moreover, by determining which of these real Lie algebras consist of Hamiltonian vector fields with respect to a Poisson structure, we provide the complete local classification of Lie-Hamilton systems on the plane. We present and study through our results new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complex Riccati and Buchdahl equations as well as some Lotka-Volterra and nonlinear biomathematical models are analysed from this Lie-Hamilton approach.

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finitedimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the properties of such Lie algebras on the plane, e.g., their associated Poisson bivectors. We study new and known Lie-Hamilton systems on R 2 with physical, biological and mathematical applications. New results cover Cayley-Klein Riccati equations, the here defined planar diffusion Riccati systems, complex Bernoulli differential equations and projective Schrödinger equations. Constants of motion for planar Lie-Hamilton systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach. MSC: 34A26 (primary) 70G45, 70H99 (secondary)Hence, system (1.4) with a R 1 (t) = 0 is a LH system as it is related to a t-dependent vector field taking values in a Vessiot-Guldberg Lie algebra V of Hamiltonian vector fields relative to Λ. Since V ≃ R ⋉ R 2 ≃ X 1 ⋉ X 2 , X 3 , X 1 ∧ X 2 = 0 and ad X 1 : X i ∈ X 2 , X 3 → [X 1 , X i ] ∈ X 2 , X 3 is not diagonalizable over R, we see in view of table 1 that the Lie algebra V belongs to class P 1 and V ≃ iso(2). Meanwhile, the LH algebra spanned by h 1 , h 2 , h 3 , h 0 is isomorphic to the centrally extended Euclidean algebra iso(2) (see also [13] for further details).On the other hand, we recall that superposition rules for LH systems can be obtained in an algebraic way by applying a Poisson coalgebra approach [19]. In contrast, other methods to derive superposition rules require to integrate a Vessiot-Guldberg Lie algebra, e.g., the group theoretical method [2], or to solve a family of PDEs [9]. Winternitz and coworkers have also derived superposition rules for Lie systems in particular forms [2,5]. The application of this latter result for general Lie systems requires to map them into the canonical form for which the superposition rule was obtained. The coalgebra procedure makes these transformations unnecessary in many cases.The structure of the paper is as follows. In section 2 we summarize the local classification of Vessiot-Guldberg Lie algebras as of Hamiltonian vector fields on the plane performed in [13], where the corresponding symplectic structures were derived by solving a system of PDEs. As a first new achievement, we show in section 3 that such symplectic structures can be determined through algebraic and geometric methods. Although our techniques are heavily based upon the Lie algebra structure of Vessiot-Guldberg Lie algebras, they also depend on their geometric properties as Lie algebras of vector fields. In particular, a new method to construct symplectic structures for LH systems on the plane related to non-simple Vessiot-Guldberg Lie algebras is described.Next we remark that a Vessiot-Guldberg Lie algebra on the plane isomorphic to sl(2) can be diffeomorphic to either P 2 , I 3 ...

The coalgebra approach to the construction of classical integrable systems from Poisson coalgebras is reviewed, and the essential role played by symplectic realizations in this framework is emphasized. Many examples of Hamiltonians with either undeformed or q-deformed coalgebra symmetry are given, and their Liouville superintegrability is discussed. Among them, (quasi-maximally) superintegrable systems on N -dimensional curved spaces of nonconstant curvature are analysed in detail. Further generalizations of the coalgebra approach that make use of comodule and loop algebras are presented. The generalization of such a coalgebra symmetry framework to quantum mechanical systems is straightforward.

The construction of N-dimensional (ND) integrable systems from coalgebras is reviewed. In the case of Poisson coalgebras, a necessary condition for the integrability of the ND coalgebra Hamiltonian coming from a given coalgebra is obtained in terms of the dimension of the symplectic realization and the number of nonlinear Casimir functions. From this viewpoint, the full set of three-, four- and five-dimensional Lie–Poisson coalgebras is analysed, and many new families of multiparametric ND integrable systems coming from the cases that fulfil the integrability condition are obtained, including the explicit form of the integrals of the motion. The superintegrability of these Hamiltonians is also emphasized, and the generalization of the whole construction to the quantum mechanical case is straightforward.

The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real three-dimensional Poisson-Lie group. As a consequence, the Poisson coalgebra map ∆ (2) that is given by the group multiplication provides the keystone for the explicit construction of a new family of 3N-dimensional integrable systems that, under certain constraints, contain N sets of deformed versions of the 3D LV equations. Moreover, by considering the most generic Poisson-Lie structure on this group, a new two-parametric integrable perturbation of the 3D LV system through polynomial and rational perturbation terms is explicitly found.