in [@ibr07d]. I introduce here the general concept of *nonlinear self-adjointness*. It embraces the previous notions of self-adjoint and quasi self-adjoint equations and includes the linear self-adjointness as a particular case. But the set of nonlinearly self-adjoint equations is essentially wider and includes, in particular, *all* linear equations. The construction of conservation laws demonstrates a practical significance of the nonlinear self-adjointness. Namely, *conservation laws can be associated with symmetries for all nonlinearly self-adjoint differential equations and systems*. In particular, this is possible for all linear equations and systems.

Notation {#nota}
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We will use the following notation. The independent variables are denoted by $$x = (x^1, \ldots, x^n).$$ The dependent variables are $$u = (u^1, \ldots, u^m).$$ They are used together with their first-order partial derivatives $u_{(1)}:$ $$u_{(1)} = \{u^\alpha_i\}, \quad u^\alpha_i = D_i (u^\alpha),$$ and higher-order derivatives $u_{(2)}, \ldots, u_{(s)}, \ldots,$ where $$u_{(2)} = \{u^\alpha_{ij}\}, \quad u^\alpha_{ij} = D_i D_j
  (u^\alpha),\ldots,$$ $$u_{(s)} = \{u^\alpha_{i_1\cdots i_s}\}, \quad u^\alpha_{i_1\cdots i_s}
  = D_{i_1} \cdots D_{i_s}(u^\alpha).$$ Here $D_i$ is the total differentiation with respect to $x^i:$ $$\label{sa:cl.diff1}
  D_i = \frac{\partial}{\partial x^i} +
u^\alpha_{i}\frac{\partial}{\partial u^\alpha} +
u^\alpha_{ij}\frac{\partial}{\partial u^\alpha_j} +
 \cdots\,.$$

A locally analytic function $f(x, u, u_{(1)}, \ldots, u_{(k)})$ of any finite number of 