LVM Encrypted Partition without Partition Table

I have the following Problem. My Laptop is fully encrypted with LVM I know the Passphrase, but the partition table is broken. I booted from an USB Stick in this Case sdb1. Is there any hope to get /dev/sda2 working again? sda1 ist mountable If I try to boot I will fall into initframs with no Volumne Groups found.

System is a Linux Mint Qiana.

 sudo fdisk -l

Disk /dev/sda: 500.1 GB, 500107862016 bytes
255 heads, 63 sectors/track, 60801 cylinders, total 976773168 sectors
Units = sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 4096 bytes
I/O size (minimum/optimal): 4096 bytes / 4096 bytes
Disk identifier: 0x000ca01e

   Device Boot      Start         End      Blocks   Id  System
/dev/sda1   *        2048      499711      248832   83  Linux
/dev/sda2          501758   976771071   488134657    5  Extended
Partition 2 does not start on physical sector boundary.

Disk /dev/sdb: 31.9 GB, 31910264832 bytes
255 heads, 63 sectors/track, 3879 cylinders, total 62324736 sectors
Units = sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0x00023cc3

   Device Boot      Start         End      Blocks   Id  System
/dev/sdb1   *        2048    62324735    31161344    c  W95 FAT32 (LBA)

Thanks for any help

EDIT 1: after Analyse and before deeper search

Sat Jan  3 11:02:41 2015
Command line: TestDisk

TestDisk 6.14, Data Recovery Utility, July 2013
Christophe GRENIER <grenier@cgsecurity.org>
http://ift.tt/TT2Ane
OS: Linux, kernel 3.13.0-24-generic (#46-Ubuntu SMP Thu Apr 10 19:11:08 UTC 2014) x86_64
Compiler: GCC 4.8
Compilation date: 2013-10-29T01:29:29
ext2fs lib: 1.42.9, ntfs lib: libntfs-3g, reiserfs lib: none, ewf lib: none
/dev/sda: LBA, HPA, LBA48, DCO support
/dev/sda: size       976773168 sectors
/dev/sda: user_max   976773168 sectors
/dev/sda: native_max 976773168 sectors
/dev/sda: dco        976773168 sectors
Warning: can't get size for Disk /dev/mapper/control - 0 B - 1 sectors, sector size=512
Hard disk list
Disk /dev/sda - 500 GB / 465 GiB - CHS 60801 255 63, sector size=512 - ST500LT012-9WS142, S/N:W0VA4SGP, FW:0001LVM1
Disk /dev/sdb - 31 GB / 29 GiB - CHS 30432 64 32, sector size=512 - Multiple Card  Reader, FW:1.00
Disk /dev/sdd - 2000 GB / 1863 GiB - CHS 30400 255 63, sector size=4096 - Seagate Expansion Desk, FW:0604

Partition table type (auto): Intel
Disk /dev/sda - 500 GB / 465 GiB - ST500LT012-9WS142
Partition table type: Intel

Analyse Disk /dev/sda - 500 GB / 465 GiB - CHS 60801 255 63
Geometry from i386 MBR: head=255 sector=63
Current partition structure:
 1 * Linux                    0  32 33    31  26 59     497664
 2 E extended                31  59 27 60801  47 46  976269314

search_part()
Disk /dev/sda - 500 GB / 465 GiB - CHS 60801 255 63

recover_EXT2: s_block_group_nr=0/30, s_mnt_count=36/4294967295, s_blocks_per_group=8192, s_inodes_per_group=2008
recover_EXT2: s_blocksize=1024
recover_EXT2: s_blocks_count 248832
recover_EXT2: part_size 497664
     Linux                    0  32 33    31  26 59     497664
     ext2 blocksize=1024 Sparse superblock, 254 MB / 243 MiB

     Linux                   31  59 29    31 124 29       4096
     LUKS 1 (Data size unknown), 2097 KB / 2048 KiB

recover_EXT2: s_block_group_nr=0/974, s_mnt_count=1/4294967295, s_blocks_per_group=32768, s_inodes_per_group=8192
recover_EXT2: s_blocksize=4096
recover_EXT2: s_blocks_count 31929088
recover_EXT2: part_size 255432704
     Linux                43874 158 61 59774 146 20  255432704 [kali]
     ext4 blocksize=4096 Large file Sparse superblock, 130 GB / 121 GiB

Results
   * Linux                    0  32 33    31  26 59     497664
     ext2 blocksize=1024 Sparse superblock, 254 MB / 243 MiB
   P Linux                   31  59 29    31 124 29       4096
     LUKS 1 (Data size unknown), 2097 KB / 2048 KiB
   P Linux                43874 158 61 59774 146 20  255432704 [kali]
     ext4 blocksize=4096 Large file Sparse superblock, 130 GB / 121 GiB

interface_write()
 1 * Linux                    0  32 33    31  26 59     497664
 2 P Linux                   31  59 29    31 124 29       4096
 3 P Linux                43874 158 61 59774 146 20  255432704 [kali]

I don't remember that I haved kali installed at any time ;-)