Restricting his considerations to the Euclidean plane, the author shows a method leading to the solution of the equivalence problem for all Lie groups of motions. Further, he presents all transitive one-parametric system of motions in the Euclidean plane.

Several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result clarifying this effect in the case when the structure is associated to a vector distribution, is proposed.

For natural numbers $r$ and $n\ge 2$ a complete classification of natural affinors on the natural bundle ${\left(J^r{T}^{*}\right)}^{*}$ dual to $r$-jet prolongation $J^r{T}^{*}$ of the cotangent bundle over $n$-manifolds is given.

For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T *(J (r,s,q)(Y, R 1,1)0)*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T *(J (r,s)(Y, R)0)*→R for any Y as above.

For natural numbers $r$ and $n$ and a real number $a$ we construct a natural vector bundle $T^{\left(r\right),a}$ over $n$-manifolds such that $T^{\left(r\right),0}$ is the (classical) vector tangent bundle $T^{\left(r\right)}$ of order $r$. For integers $r\ge 1$ and $n\ge 3$ and a real number $a<0$ we classify all natural operators $T_{|M_n}⇝T{T}^{\left(r\right),a}$ lifting vector fields from $n$-manifolds to $T^{\left(r\right),a}$.

For integers $r\ge 2$ and $n\ge 2$ a complete classification of all natural operators $A:T_{|M_n}⇝T{\left(J^r{T}^{*}\right)}^{*}$ lifting vector fields to vector fields on the natural bundle ${\left(J^r{T}^{*}\right)}^{*}$ dual to $r$-jet prolongation $J^r{T}^{*}$ of the cotangent bundle over $n$-manifolds is given.

All natural operators T ↝ T(T ⊗ T*) lifting vector fields X from n-dimensional manifolds M to vector fields B(X) on the bundle of affinors ™ ⊗ T*M are described.

We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor $A\left(f\right):T{T}^{\left(r\right)}M\to T{T}^{\left(r\right)}M$ on the vector r-tangent bundle $T^{\left(r\right)}M=\left(J^r(M,ℝ)₀\right)*$ over M. This problem is reflected in the concept of natural operators $A:T_{|ℳfₙ}^{(0,0)}⇝T^{(1,1)}{T}^{\left(r\right)}$. For integers r ≥ 1 and n ≥ 2 we prove that the space of all such operators is a free (r+1)²-dimensional module over ${}^{\infty }\left(T^{\left(r\right)}ℝ\right)$ and we construct explicitly a basis of this module.

For natural numbers $r\ge 2$ and $n$ a complete classification of natural transformations $A:T{T}^{\left(r\right)}\to T{T}^{\left(r\right)}$ over $n$-manifolds is given, where $T^{\left(r\right)}$ is the linear $r$-tangent bundle functor.

The notion of a structure tensor of section of first order natural bundles with homogeneous standard fibre is introduced. Properties of the structure tensor operator are studied. The universal factorization property of the structure tensor operator is proved and used for classification of first order $*$-natural differential operators $\underline{D}:\underline{T\times T}\to \underline{T}$ for $n\ge 3$.

In this paper we describe the close relationship between invariant evolutions of projective curves and the Hamiltonian evolutions of Adler, Gel'fand and Dikii. We also show how KdV evolutions are related as well to invariant evolutions of projective surfaces.