vada

 https://www.google.com/search?q=Avataravada

Four Children

1. first son Nivritidev was the incarnation of Shiva, 
2. second son Gyaneshwar was the incarnation of Vishnu and 
3. third son Sopandev was the incarnation of Brahma, 
4. fourth child Muktha Bai was the incarnation of Shakthi.

https://harekrishnainfo.blogspot.com/search/label/VitthalRukmini

karuna sindhu

 https://www.google.com/search?q=karuna+sindhu

First

 https://www.google.com/search?q=First+Avatar

Mahishasura Mahishi

1. The three Trinity forces combined and Dattathreya was born. 
2. The three Trinity consort forces combined and was born as daughter to Rishi Galava. She was named Leelavathi. 
3. Dattathreya and Leelavathi joined in wed-lock and they lived happily for a good length of time. 

https://freeglobaluniversity.blogspot.com/search/label/Trinity

 As time went on Dattathreya yearned to go back and merge in Trinity. Leelavathi would not let him go. There was an altercation, at the end of which 

1. Leelavathi cursed Dattathreya born as a Mahishan (He-buffalo). 
2. Dattathreya cursed Leelavathi be born a Mahishi (She-buffalo). 

The two curses took effect, Leelavathi becoming a Mahishi and Dhattatreya becoming a Mahishan. The two (Dattathreva and Leelavathi) were reveling together for sometime.

https://www.google.com/search?site=&tbm=isch&source=hp&biw=1600&bih=770&q=Mahishasura+Mahishi
http://www.slideshare.net/search/slideshow?searchfrom=header&q=Mahishasura+Mahishi
http://www.youtube.com/results?search_query=Mahishasura+Mahishi 

The Curse Was Broken by Liberation given by Each Other

1. Male Mahishasura  by Female Trinity Goddess ( Adi Shakti Saraswati Laxmi Parvathi)
2. Female Mahishi by Male Trinity God (Brahmachari(Brahma) Hari(Vishnu) Hara (Shiva)

https://k00y.blogspot.com/search/label/Mahishasura

https://freeglobaluniversity.blogspot.com/search/label/Ayyappan

avaradar

 https://www.youtube.com/user/avaradar/videos

Half hanuman half ganesh

https://www.google.com/search?site=&tbm=isch&source=hp&biw=1600&bih=721&q=half+hanuman+half+ganesh
https://www.youtube.com/results?search_query=half+hanuman+half+ganesh

Common Offering of Laddu - Favourite of Both Deities
https://www.youtube.com/results?q=How+to+Make+Laddu

https://ch1k.blogspot.com/search/label/eat

Both Form Entrance of Most Swami Narayan Temples
http://n1r1.blogspot.com/

The idol is an amalgam of Sri vigneshwara and Sri Anjaneya on the lines of

Ardhanareeswara ( Shiva and Parvati) and 
http://freeglobaluniversity.blogspot.com/search/label/HalfMaleFemale

Sri Sankaranarayana (combination of Siva and narayana).
http://freeglobaluniversity.blogspot.com/search/label/FormFormlessGOD

Sri Vigneswara and Sri Anjaneya are only two deities who are totally free from any influence over them by the Navagrahas.

https://freeglobaluniversity.blogspot.com/search/label/Planet

Both could not identify their beloved ones at first meet.(ganesh-shiva)(hanuman-sri ram)
http://bu08.blogspot.com/search/label/HanumanGanesh