Abstract:
We establish a new class of integrable {\it systems of Kowalevski type}, associated with discriminantly separable polynomials of degree two in each of three variables. Defining property of such polynomials, that all discriminants as polynomials of two variables are factorized as products of two polynomials of one variable each (denote one of the polynomial components as $P$), lead to an effective integration procedure. In the motivating example, the celebrated Kowalevski top, the discriminant separability is a property of the polynomial defining the Kowalevski fundamental equation. We construct several new examples of systems of Kowalevski type, and we perform their explicit integration in genus two theta-functions. One of the main tasks of the paper is to classify such discriminantly separable polynomials. Our classification is based on the study of structures of zeros of a polynomial component $P$ of a discriminant. From a geometric point of view, such a classification is related to the types of pencils of conics. We construct also discrete integrable systems on quad-graphs associated with discriminantly separable polynomials. We establish a relationship between our classification and the classification of integrable quad-graphs which has been suggested recently by Adler, Bobenko and Suris. As a fit back, we get a geometric interpretation of their results in terms of pencils of conics, and in the case of general position, when all four zeros of the polynomial $P$ are distinct, we get a connection with the Buchstaber-Novikov two-valued groups on $\mathbb {CP}^1$.

Abstract:
Starting from the notion of discriminantly separable polynomials of degree two in each of three variables, we construct a class of integrable dynamical systems. These systems can be integrated explicitly in genus two theta-functions in a procedure which is similar to the classical one for the Kowalevski top. The discriminnatly separable polynomials play the role of the Kowalevski fundamental equation. The natural examples include the Sokolov systems and the Jurdjevic elasticae.

Abstract:
The contributions of Sophya Kowalewski to the integrability theory of the equations for the heavy top extend to a larger class of Hamiltonian systems on Lie groups; this paper explains these extensions, and along the way reveals further geometric significance of her work in the theory of elliptic curves. Specifically, in this paper we shall be concerned with the solutions of the following differential system in six variables h_1,h_2,h_3,H_1,H_2,H_3 dH_1/dt = H_2 H_3 (1/c_3 - 1/c_2) + h_2 a_3 - h_3 a_2, dH_2/dt = H_1 H_3 (1/c_1 - 1/c_3) + h_3 a_1 - h_1 a_3, dH_3/dt = H_1 H_2 (1/c_2 - 1/c_1) + h_1 a_2 - h_2 a_1, dh_1/dt = h_2 H_3/c_3 - h_3 H_2/c_2 + k (H_2 a_3 - H_3 a_2), dh_2/dt = h_3 H_1/c_1 - h_1 H_3/c_3 + k (H_3 a_1 - H_1 a_3), dh_3/dt = h_1 H_2/c_2 - h_2 H_1/c_1 + k (H_1 a_2 - H_2 a_1), in which a_1,a_2,a_3,c_1,c_2,c_3 and k are constants.

Abstract:
In this work, we give sufficient conditions in order to have finite ramification locus in sequences of function fields defined by different kind of Kummer extensions. These conditions can be easily implemented in a computer to generate several examples. We present some new examples of asymptotically good towers of Kummer type and we show that many known examples can be obtained from our general results.

Abstract:
We construct higher-dimensional generalizations of the classical Hess-Appel'rot rigid body system. We give a Lax pair with a spectral parameter leading to an algebro-geometric integration of this new class of systems, which is closely related to the integration of the Lagrange bitop performed by us recently and uses Mumford relation for theta divisors of double unramified coverings. Based on the basic properties satisfied by such a class of systems related to bi-Poisson structure, quasi-homogeneity, and conditions on the Kowalevski exponents, we suggest an axiomatic approach leading to what we call the "class of systems of Hess-Appel'rot type".

Abstract:
Deformations of the known polynomial Poisson pencils associated with the Kowalevski top are studied. As a result we find new variables of separation from the one of the Yehia systems and new bi-Hamiltonian description of the integrable deformation of the Kowalevski gyrostat in two fields proposed by Sokolov and Tsiganov.

Abstract:
A new view on the Kowalevski top and the Kowalevski integration procedure is presented. For more than a century, the Kowalevski 1889 case, attracts full attention of a wide community as the highlight of the classical theory of integrable systems. Despite hundreds of papers on the subject, the Kowalevski integration is still understood as a magic recipe, an unbelievable sequence of skilful tricks, unexpected identities and smart changes of variables. The novelty of our present approach is based on our four observations. The first one is that the so-called fundamental Kowalevski equation is an instance of a pencil equation of the theory of conics which leads us to a new geometric interpretation of the Kowalevski variables $w, x_1, x_2$ as the pencil parameter and the Darboux coordinates, respectively. The second is observation of the key algebraic property of the pencil equation which is followed by introduction and study of a new class of {\bf discriminantly separable polynomials}. All steps of the Kowalevski integration procedure are now derived as easy and transparent logical consequences of our theory of discriminantly separable polynomials. The third observation connects the Kowalevski integration and the pencil equation with the theory of multi-valued groups. The Kowalevski change of variables is now recognized as an example of a two-valued group operation and its action. The final observation is surprising equivalence of the associativity of the two-valued group operation and its action to $n=3$ case of the Great Poncelet Theorem for pencils of conics.

Abstract:
The phase topology of the integrable Hamiltonian system on $e(3)$ found by V.V.Sokolov (2001) and generalizing the Kowalevski case is investigated. The generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. Relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the classification of iso-energy manifolds of the reduced systems with two degrees of freedom is given. The set of critical points of the complete momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the surfaces bearing the bifurcation diagram of the momentum map. We give examples of the existing iso-energy diagrams with a complete description of the corresponding rough topology (of the regular Liouville tori and their bifurcations).

Abstract:
Surfaces of general type with geometric genus $p_g=0$, which can be given as Galois covering of the projective plane branched over an arrangement of lines with Galois group $G=(\mathbb Z/q\mathbb Z)^k$, where $k\geq 2$ and $q$ is a prime number, are investigated. The classical Godeaux surface, Campedelli surfaces, Burniat surfaces, and a new surface $X$ with $K_X^2=6$ and $(\mathbb Z/3\mathbb Z)^3\subset {Tors} (X)$ can be obtained as such coverings. It is proved that the group of automorphisms of a generic surface of the Campedelli type is isomorphic to $(\mathbb Z/2\mathbb Z)^3$. The irreducible components of the moduli space containing the Burniat surfaces are described. It is shown that the Burniat surface $S$ with $K_S^2=2$ has the torsion group ${Tors} (S)\simeq (\mathbb Z/2\mathbb Z)^3$, (therefore, it belongs to the family of the Campedelli surfaces), i.e., the corresponding statement in the papers of C. Peters "On certain examples of surfaces with $p_g=0$" in Nagoya Math. J. {\bf 66} (1977), and I. Dolgachev "Algebraic surfaces with $q=p_g=0$" in {\it Algebraic surfaces}, Liguori, Napoli (1977), and in the book of W. Barth, C. Peters, A. Van de Ven "Compact complex surfaces", p. 237, about the torsion group of the Burniat surface $S$ with $K_S^2=2$ is not correct.

Abstract:
It has been proved that on 2-dimensional orientable compact manifolds of genus $g>1$ there is no integrable geodesic flow with an integral polynomial in momenta. There is a conjecture that all integrable geodesic flows on $T^2$ possess an integral quadratic in momenta. All geodesic flows on $S^2$ and $T^2$ possessing integrals linear and quadratic in momenta have been described by Kolokol'tsov, Babenko and Nekhoroshev. So far there has been known only one example of conservative system on $S^2$ possessing an integral cubic in momenta: the case of Goryachev-Chaplygin in the dynamics of a rigid body. The aim of this paper is to propose a new one-parameter family of examples of complete integrable conservative systems on $S^2$ possessing an integral cubic in momenta. We show that our family does not include the case of Goryachev-Chaplygin.