I. INTRODUCTION A. Nonlinear reaction-diffusion equations A. Nonlocal symmetries of nonlinear diffusionequations 1. The case when

*Q*(

*u*) is arbitrary 1. Inverse potential system arising from

**Y**

_{1}2. Inverse potential system arising from

**X**

_{3}when

*Q*(

*u*) =

*u*

^{3}3. Inverse potential system arising from

**X**

_{4}when

*Q*(

*u*) =

*e*

^{ u }4. The case when

*Q*(

*u*) =

*u*ln

*u*1. Inverse potential system arising from

**Y**

_{1}2. Inverse potential system arising from

**X**

_{3}when

*Q*(

*u*) =

*u*

^{3}3. Inverse potential system arising from

**X**

_{4}when

*Q*(

*u*) =

*e*

^{ u }4. The case when

*Q*(

*u*) =

*u*ln

*u*II. NEW METHOD: NONLOCALLY RELATED PDESYSTEMS ARISING FROM POINT SYMMETRIES 1. Inverse potential system arising from

**Y**

_{1}2. Inverse potential system arising from

**X**

_{3}when

*Q*(

*u*) =

*u*

^{3}B. Nonlinear diffusionequations B. Nonlocal symmetries of nonlinear waveequations 1. Inverse potential system arising from

**Y**

_{1}2. Inverse potential system arising from

**X**

_{3}when

*Q*(

*u*) =

*u*

^{3}2. Inverse potential system arising from

**Y**

_{2}3. Inverse potential system arising from

**Y**

_{3}4. Inverse potential system arising from

**Y**

_{4}5. Inverse potential system for a nonlinear diffusionequation(3.44) 3. Inverse potential system arising from

**Y**

_{3}4. Inverse potential system arising from

**Y**

_{4}5. Inverse potential system for a nonlinear diffusionequation(3.44) III. EXAMPLES OF INVERSE POTENTIAL SYSTEMS A. Nonlinear reaction-diffusion equations 1. The case when

*Q*(

*u*) is arbitrary 1. Inverse potential system arising from

**Y**

_{1}2. Inverse potential system arising from

**X**

_{3}when

*Q*(

*u*) =

*u*

^{3}3. Inverse potential system arising from

**X**

_{4}when

*Q*(

*u*) =

*e*

^{ u }4. The case when

*Q*(

*u*) =

*u*ln

*u*B. Nonlinear diffusionequations 1. Inverse potential system arising from

**Y**

_{1}2. Inverse potential system arising from

**X**

_{3}when

*Q*(

*u*) =

*u*

^{3}2. Inverse potential system arising from

**Y**

_{2}3. Inverse potential system arising from

**Y**

_{3}4. Inverse potential system arising from

**Y**

_{4}5. Inverse potential system for a nonlinear diffusionequation(3.44) C. Nonlinear waveequations 3. Inverse potential system arising from

**X**

_{4}when

*Q*(

*u*) =

*e*

^{ u }3. Inverse potential system arising from

**Y**

_{3}4. The case when

*Q*(

*u*) =

*u*ln

*u*4. Inverse potential system arising from

**Y**

_{4}5. Inverse potential system for a nonlinear diffusionequation(3.44) IV. EXAMPLES OF NONLOCAL SYMMETRIES ARISING FROM THE SYMMETRY-BASED METHOD A. Nonlocal symmetries of nonlinear diffusionequations 4. The case when

*Q*(

*u*) =

*u*ln

*u*B. Nonlocal symmetries of nonlinear waveequations 4. Inverse potential system arising from

**Y**

_{4}4. The case when

*Q*(

*u*) =

*u*ln

*u*4. Inverse potential system arising from

**Y**

_{4}V. CONCLUSION AND OPEN PROBLEMS 5. Inverse potential system for a nonlinear diffusionequation(3.44) 5. Inverse potential system for a nonlinear diffusionequation(3.44)

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