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We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansätze reducing nonlinear evolution equations to system of ordinary differential equations. The ansätze are constructed by using operators of nonpoint classical and conditional symmetry. Then we find solution to nonlinear heat equation which cannot be obtained in the framework of the classical Lie approach. By using operators of Lie-Bäcklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too.

It is well known that the classical Lie symmetry method of point transformations is often used for reducing the number of independent variables in partial differential equation to obtain ordinary differential equations. After integration of reduced differential equations one can obtain partial solutions of the equation under study [

We use operators of nonpoint classical and conditional symmetries to extend the class of differential equations to which the symmetry method is applicable. In this paper we study the symmetry reduction of partial differential equations with two independent variables by using the operators of nonpoint symmetry because the prolongated operators of classical point symmetry lead to the classical invariant solutions. The method can be naturally generalized to the multidimensional case. We construct the ansatz for dependent variable

Recall, that the well-known integrable nonlinear differential equations such as Korteweg-de-Vries, sine-Gordon, and cubic Schroedinger equations admit an infinite number of Lie-Bäcklund symmetry operators [

The concept of differential invariant solutions based on infinite Lie group

We search for the ansatz for the derivatives of such form

The tangent transformations groups are also used in the framework of this approach. Let us consider the nonlinear evolution equation

Next we emphasize that the operators of conditional symmetry of corresponding system can be used for construction the Bäcklund transformations for nonlinear wave equation

Note that this approach is also applicable for linearization of nonlinear partial differential equations with two independent variables. Indeed, consider the second-order differential equation

Let us note that the symmetry group of corresponding system written in the general form contains the symmetry group of point transformations of initial equation as a subgroup and generators of point transformations can be used to construct ansatz (

Further we show how the operators of Lie-Bäcklund symmetry [

We use Theorem 1 from [

Substituting (

We have constructed ansätze (

As was noted above the linearization of class of nonlinear partial differential equations (

Finally we show that the existence of even at least one operator of Lie-Bäcklund symmetry to ordinary differential equations (

The authors declare that there is no conflict of interests regarding the publication of this paper.